Ion Cyclotron (IC) Oscillations Excited by Nonlinear Waves Propagating in Collision-free Auroral Ionosphere

We study ion cyclotron (IC) oscillations activated by a stochastic, strong space-charge electric wavefield E of nonlinear waves propagating auroral ionosphere. E is in a plane perpendicular to the ambient magnetic field B. The word “strong” means that (1) the conventional linear plasma wave model connected to a perturbed electric field is not suitable to be employed; (2) the E × B drift is comparable to (even higher than) the thermal speed of particles, and drive them away from the initial thermal equilibrium. A physical model is set up for a dense cluster of electron soliton trains with which a magnetic flux tube is teeming. Then, the collision-free Boltzmann equation is solved under the condition that E is temporally constant. With a nonzero initial guiding-center (GC) velocity, ions are found to follow a double-circle trajectory in velocity space with an IC oscillation frequency ω which shifts from the magnetic gyrofrequency Ω = eB/mi (where e and mi are the *Corresponding author: E-mail: zma@mymail.ciis.edu Ma and St.-Maurice; PSIJ, 12(3), 1-22, 2016; Article no.PSIJ.29209 charge and mass of the singly ionized ions, respectively). Furthermore, the “constant” condition is relaxed by using a simple stochastic E which has 10-step random strengths in 10 different time intervals. The accommodation of ω (as well as other parameters) is illustrated in response to the E switches. At last, the work is generalized by using two random-number generators for the strength and time, respectively. In this case, ω can be shifted to several Ω values. This result is in good agreement with what FAST satellite measured in auroral field-aligned current regions.


INTRODUCTION
Electrostatic ion-cyclotron (IC) oscillation mode was firstly predicted by Stix [1] in studying an infinitely long, cylindrically symmetric plasma column of finite density (n0) at zero pressure immersed in a uniform axial magnetic field (B). He pointed out that when ck ≫ ωUH and ck/ω ≫ ωUH /Ω [where c is the speed of light, k the wave vector along B, ω the wave frequency, Ω = (1/c)eB/mi the ion cyclotron frequency with (e, mi) the charge and mass of the singly ionized ion, respectively, ωUH = √ ω 2 pi + Ω 2 the ion upper-hybrid frequency where ωpi = √ 4πne 2 /mi is the ion plasma frequency], the extraordinary hydromagnetic wave becomes a wave whose natural frequency approaches Ω. He later found that when an IC wave propagates in a plasma along a weakening B, ω becomes enhanced; in the vicinity of ω = Ω, wave energy can be absorbed by the plasma via cyclotron damping with extremely efficient power transfers [2]. At the same time, Bernstein [3] studied extensively electron and ion oscillations of a fully ionized, collision-free plasma in a static external magnetic field. For the low-frequency ion oscillations, he predicted two modes at cyclotron harmonics: longitudinal ion waves and transverse hydromagnetic waves.
Stix's effective plasma heating mechanism drew much attention in theoretical and laboratory plasma studies.
The energization process was soon confirmed by experiments on a magnetic mirror device called the B-66 machine [4]. Besides, another experiment performed in cesium and potassium plasmas also observed the IC oscillations when the electron drift was ∼10 times the ion thermal velocity, but the frequency was slightly higher than Ω [5]. Drummond & Rosenbluth [6] explained that the IC oscillations was excited by the magnetic fieldaligned currents (FACs); and Woods [7] extended Stix's work by taking into consideration the effects of the plasma viscosity and compressibility. By using a generalized dispersion relation, the author showed that a range of oscillation frequencies is possible which are well beyond Ω. Furthermore, Yoshikawa et al. [8] showed that the ion heating is also contributed by the electron Landau damping and electron-ion collisions. Later, Hosea & Sinclair [9,10] exposed that even the IC wave propagation in a plasma is influenced by the electron inertia. In an experiment where torsional Alfven waves were excited, Müller [11] examined the coupling of ions with neutrals with a decreasing ionization degree. He witnessed that both k and the damping effect peaked at Ω.
Interestingly, the IC-resonance acceleration principle was soon applied to separate ion isotopes and ions with different charge-to-mass ratios in plasmas [12,13].
But if a plasma contains only two ion species, Sawley & Tran [14] noticed that the IC frequency lies approximately midway between the two Ω values. If more than one ion species, IC modes are strongly influenced by electron dynamics (the electron inertia and Landau damping) in a low density plasma cavity, just as revealed in the plasma with one type of ions [8,[15][16][17]. Another important result was obtained by Ono et al. [18]. The authors measured profiles of wave absorptions versus ion temperatures in an ACT-1 hydrogen plasma. They identified that the excellent efficient IC resonant heating occurs near the fifth IC harmonics of deuterium-like and tritium-like ions. It deserves to mention an intriguing experiment designed by Sato & Hatakeyama [19]. The authors used a voltage-biased electrode to drive an IC oscillation. The potential penetrates into a plasma channel parallel to B in front of the electrode, whileas the penetration is limited by the radial escape of ions from the plasma channel. This radial escape was found to obey the cyclotron motion with a period of Ω in the region close to the electrode.
In a plasma containing both negative and positive ions, D'Angelo & Merlino [20] and Song et al. [21] found that there are two branches of the IC modes for mi+ > mi−: One is the "lowfrequency" mode ω ≥ Ω+, and the other is the "high-frequency" one ω ≥ Ω−. The frequencies were found to increase with the increasing percentage of negative ions. Particularly, by employing the standard linear Vlasov theory, Chow & Rosenberg [22,23] investigated the effects of heavier dust and heavier negative ions on the collisionless IC instabilities, respectively. In the former case, the authors showed that positively charged dust tends to stabilize IC waves, while negatively charged dust facilitates triggering the instability; in the latter case, by contrast, the instabilities of both the light and heavy ions are easier to be excited when more charge is carried by negative ions. In both cases, the critical electron drifts to excite the instabilities decrease when the relative density of negative ions increases. In another dusty plasma experiment to observe the IC waves, Barkan et al. [24] verified that negatively charged dust makes the plasma more unstable. The newest experimental work was done by Kim et al. [25]. They used very heavy negative ions (C7F − 14 ) in the plasma and observed that these ions increase both the number and the intensity of the excited harmonic IC modes.
In geospace, especially in auroral regions, electrostatic IC oscillations have been detected by rockets and satellites. Early space-borne measurements of harmonic IC oscillations in the VLF bands were done at altitudes <2700 km [26][27][28][29]. Following these measurements, numerous vehicles diagnosed IC oscillations during last decades, such as S3-3 [30], DE-1 [31,32], ISEE-1 [33], Polar [34], FAST [35], and Cluster [36]. Because many observations provided evidence of parallel electric fields and/or field-aligned currents existing in regions where IC waves were measured, the excitation of the IC waves was naturally connected to the current-driven mechanism as considered by laboratory experiments and theoretical studies (e.g., [37]). It was suggested that if the fieldaligned currents increase to a threshold, say, tens of µA/m 2 in strength, the IC instability [and/or the Buneman instability, the ion acoustic (IA) instability] can be triggered; the plasma turbulence and then anomalous collisions thus induced in turn moderate the currents via anomalous transport coefficients [38,39]. This process was considered to lead to two possible conspicuous consequences: one is the formation of nonwave phase-space clumps originated from the growth of the plasma instability [40], and the other is the enhanced Joule heating in plasmas due to the anomalous resistivity [41].
By excluding any nonlinear dispersion relations, viscous heating terms, and chemical reactions, Forme et al. [42] presented the criteria for the wave ignitions due to current-driven instabilities. The authors brought to light following results by using an isotropic, time-dependent model: (1) · v the is the generalized Te v thi , the IC instability begins. The heating rates take ∂Te ∂t = , and ν * i is the ion anomalous collision frequency satisfying miν * i = meν * e ]. Numerical calculations exhibited that the first two instabilities produce bigger electron heating rates due to much higher collision frequencies which inhibit the electron heat conduction processes, while the first instability contributes little to the ion temperature ratio, and the IC instability seems to be initiated at the lowest altitudes while all of them should exist above 1000 km for strong currents with tens of µA/m 2 .
Obviously, within the scope of linear plasma waves, IC oscillations can be reasonably explained by current-driven mechanism. Nevertheless, as foretold by Bernstein, Green, & Kruskal (BGK) [3] that localized non-wave phase-space structures may exist in turbulent plasmas originated from nonlinear processes, numerous studies have demonstrated that coherent, nonlinear potential structures can actually be triggered by the particle trapping due to various wave-particle interactions (see reviews in [43]). Specifically, there are three stages in the development [44]: (1) Electrons are switched from a free state to a trapped state in a potential well, leading to the formation of electron phase-space holes by instabilities; (2) The potential structures grow and deepen with the growth of electron phase-space holes; the anomalous transport properties increase to develop ion trapping; and (3) After several ion cyclotron periods, localized non-wave structures (also called solitary waves, space-charge elements, electrostatic double layers, clumps, or electrostatic shocks) are formed in phase space. If both waves and non-wave structures remain alive in electric and magnetic fields, the spacecharge structures are stratified (or "filamented") to form field-aligned "clumps" [45]; at the same time, ions response to these elements in two possible ways: they may either keep regular motions without any gains of the electromagnetic energy from the fields if their speed across the magnetic field is less than the phase velocity of a single wave [46], or, be coherently energized within a small phase space when the Dopplershifted wave frequencies are close to an integer multiple of Ω [47].
In the IC modes, Chiueh & Diamond [40] discovered that if T ⊥i ≥ T ∥i ∼ Te (where T ⊥i and T ∥i are ion temperature components perpendicular and parallel to B, respectively), the turbulence is of the wave-clump type; if T ⊥i ≫ T ∥i ∼ Te, it is the clump-dominant type. This new ingredient is space-charge "clumps", the socalled electrostatic solitons, which describes the incomplete blending of a Vlasov plasma: waveparticle interactions make stochastic orbits of particles by turbulent electric fields; the phasespace density tends to decrease to smaller scales in a finite time, and thus generate phasespace space-charge density granulations (that is, clumps); these clumps ballistically propagate at the resonant velocities v = ω − nΩi/k ∥ in a finite time. In the development, the turbulent forces produced by the turbulent electromagnetic field tend to tear the space-charge chunk of particles apart and cause the decay of the clumps. However, the size of the space-charge structures is so small that they keep every particle feels the same force. Thus, the elements retain their structural integrity for a relatively longer time than the average correlation time of the system. This effect offsets the separation tendency caused by turbulent processes.
In addition, the authors found that the shape of the space-charge structures does not rely on turbulent frequencies in position space, but does in velocity space.
In position space, for either low or high frequency turbulence, the parallel scale is of the spectrum-averaged parallel wavelengthk −1 ∥ , while the perpendicular scale is of the cylindrically-symmetric, spectrumaveraged perpendicular wavelengthk −1 ⊥ . By contrast, in velocity space, the perpendicular extent is not the same for different frequency bands: for low-frequencies, the scale can extend up to a scale dependent on the thermal velocity vT; for high frequencies, it is determined by is the gyroradius, andk ⊥ ρ ≥ 1). Specifically, the clump shape in phase space depends on the particle species. For electron clumps in the IC regime, the perpendicular diffusion dominates the parallel one withk ⊥ ≫k ∥ . So electron parallel diffusion can be ignored. Because electrons are strongly magnetized withk ⊥ ρe ≪ 1, electron clump is bounded to the magnetic field lines. In real space, it has a 1-D long cigar shape. In velocity space, it has a pancake shape of radius vte and thickness (k ∥ τe) −1 (where τe is the electron decorrelation time). The electron clump travels at a parallel speed of v ∥ ∼ nΩi/k ∥ . For ion clumps, their perpendicular diffusion is dominated bȳ k ⊥ ≫k ∥ , like electron ones. So ion parallel diffusion can also be ignored. However, ions are weakly magnetized withk ⊥ ρi ∼ 1. Ifk ⊥ ρi < 1, ion clump will show the same cigar shape as electron clump in real space. On the other hand, ifk ⊥ ρi > 1, ion clump will appear as a tether rod in real space with tether length ρi gyrating about the guiding center (GC). In both cases, the ion clump propagates at the ballistic velocity v with a radiusk −1 ⊥ and a lengthk −1 ∥ . In velocity space, ion clump appears in a pancake shape of radius vti fork ⊥ ρi < 1, and in a gyrating tether disk of radius vti(k ⊥ ρi) −1 and tether length vti fork ⊥ ρi > 1. The thickness of either the pancake or the disk is (k ∥ τi) −1 (where τi is the ion decorrelation time).
The nonlinear space-charge structures have been observed at many sites in geospace, e.g., auroral zones of mid-and-high-altitudes, bow shock, magnetotail, and solar wind due to the existence of detectable space-charge electric fields, see, e.g., [48] for detailed introductions. The nonlinear phenomena had been recognized gradually with the advance of theoretical work and observations [49]. The structures were first noticed and named as intense broadband electrostatic noises (BENs) in 1970s in the magnetotail [50,51], along with data of magnetic noise bursts (relating to cross-tail current and intervals of tailward flows), electrostatic electron cyclotron waves, and upper hybrid waves [51,52]. BEN is bursty, extending from the lowest frequencies up to as high as the plasma frequency (electron cyclotron frequency) while the intensity decreases when the frequency increases. It peaks at or below the LH frequency. During 1980s, the first space-charge structure was identified from S3-3 waveform electric-field data in the auroral acceleration region [53]. It is characterized by locally density-depleted (called "ion hole"), electric-field fluctuations in a size of Debye-scale. Because no spectral form of data were used, no link was made between these new definitions and BEN, but simply the FFT-rendering of solitary waves. Yet, theoretical investigations pointed out that certain kinds of nonlinear structures could explain the broad frequency spectra [54], and, electron acoustic solitons passing by a satellite would generate spectra that could explain the high frequency part of BEN [55]. The breakthrough eventually came when a sophisticated waveform receivers was used, which has a high temporal resolution. In 1994, Matsumoto et al. [56] analyzed the distant magnetotail with Geotail data and proved that solitary waves are as a matter of fact BENs: the measured waveform electric field of BENs is nothing but the spectra from of the electrostatic solitary waves carrying spacecharge electric fields. From that time, extensive and detailed pictures for these nonlinear electric field structures were reported from almost every high-resolution space project (see details in [57]).
It is now clear that the electrostatic solitary waves are solitary structures that behave as are spacecharge carriers to contribute strong transverse electric fields to space plasmas. Due to the presence of these fields, the characteristics of charged particles residing in the vicinity of the regions teemed with solitary waves will surely be altered. For example, in his pioneer work, Stix [1] showed unequivocally that near the IC resonant frequency the electric field determines the behavior of charged particles. This prediction was demonstrated to be valid [58]: the virtual ion cyclotron frequency ω should satisfy ω 2 = Ω 2 +Ωd(Ex/B)/dx when the first derivative of the electric field (∇ · E = dEx/dx) is nonzero, while the second derivative ( is zero. There was another example [59]: If E = E0(1 + y/L)ŷ in the cartesian geometry (where E0 is the field at y = 0, L the characteristic length, y the ion position), that is, the electric field is proportional to the distance in a specific direction in space, the authors exposed that In the cylindrical case, Ma & St.-Maurice [60] presented that where Ec is the field at r = Rc, Rc the characteristic radius, r the ion position), that is, the electric field is proportional to the radius. These studies unfolded that the frequency of the IC oscillations can shift away from Ω either positively or negatively, depending on the polarization of E perpendicular to B, but irrelevant of the origin of E. In the linear wave regime, the perturbed E is weak. The frequency shift is thus negligible. On the contrary, if E is provided by some nonlinear process, say, the space charges of solitons, the linear perturbation condition is broken. E may be so strong that the shift is unable to be overlooked. Let's use FAST data for an estimation. Ergun et al. [61] provided that the local magnetic field B is 11481 nT, and the typical peak strength of the space-charge electric field E for a soliton is 1 V/m. As a result, the local peak E × B drift (E/B) is 87 km/s. At the thermal equilibrium state of about 1000 K, the thermal speeds of protons and electrons are of the orders of 4 km/s and 170 km/s, respectively. The speed E/B is typically larger than the ion thermal speed. However, the speed E/B is small relative to the electron thermal speed. Any departures from a Maxwellian velocity distribution due to the Lorentz force cannot be negligible for ions, and we call such a electric field is "strong" for ions. In such strong fields, any traditional linear wave theory is not suitable to be used. But for thermal electrons, they can still be simply assumed to follow Maxwellian but with a E × B drift to the leading order for the situation at hand. We call such a electric field is "weak" for electrons, and traditional linear wave theories are still valid for them.
Let's use FAST data to assess the cyclotron frequency ω of ions in the strong electric field mentioned above, in order to know the order of the frequency shift. Assume that the field is cylindrically symmetric and proportional to r.
The magnetic gyrofrequency of ions is Ωi = 1100 rad/s (or 200 Hz) for a proton, and Ωe = (mi/me)Ωi for an electron. Rc scales with the ion gyroradius ρi which satisfies 2λD ≤ ρi ≤ 20λD where λD = 82 m is Debye length [62]. These parameters give 1.1Ωi ≤ ω ≤ 1.7Ωi, indicating that the IC oscillation ω always deviates from Ωi by tens of percent. For a plasma density of n0 ∼ 5.7 cm −3 , we have the ion plasma frequency ωpi is 3139 rad/s, and the electron plasma frequency ωpe is 43ωpi. Thus, the lower-hybrid (LH) frequency ωLH is 3138 rad/s (or 500 Hz). Obviously, the maximum IC oscillation frequency shifts into the LH band. This approximate evaluation explains qualitatively the FAST observations that the solitary structures are evenly spaced at a frequency above the local H + gyrofrequency (see Fig.5 in [61] for a reference). Notice that this estimation is only suitable for a single soliton case. In reality, the electric field should be produced by space charges of all soliton trains in magnetic flux tubes; and, the strength of E may not be constant with time. Therefore, the IC oscillations may have a very different picture. More detailed modeling and simulations are then needed for accurate data assimilations and quantitative explanations. Unfortunately, no relevant studies have been found to report on the IC oscillations under strong electric field conditions. This situation is understandable: traditionally, we rely on the linear plasma wave theory to consider instabilities and related oscillations, where the wavefield E is weak, behaving as a perturbation, and that measured ω should always be located at harmonics of Ω; any observable deviations from linear theory predictions are naturally attributed to the contributions of unusual initial and/or boundary conditions within the frame of the theory, regardless of the fact whether or not these conditions are still valid for the theory.
Fortunately, stochastic methods are widely used in different fields of physics [63][64][65]. Different these methods, we offer an alternative approach in this paper which illustrates the IC oscillations stimulated by a strong electric field of solitons propagating in auroral ionosphere. The electric field is produced by solitons' space charges stochastically in the plane perpendicular to the magnetic field. We focus on the ion oscillating feature, in order to see what kind of frequencyshifted IC waves can be excited. This study is totally different from those using traditional linear wave theories where the plane-wave perturbation assumption, E1 ∼ e i(k·r−ωt) , is used. For strong space-charge electric field strengths which produce large drifts to particles, this weakfield condition is broken and we have to seek for solutions with the aid of kinetic theory, and then to describe the macroscopic properties of ions in terms of their microscopic characteristics of motion. We start from setting up a physical model to describe solitons propagating in a magnetic flux tube, and give a Hamiltonian formulation for ions driven by the space-charge electric field in the tube, as given in Section 2. Then, in Section 3, we present results for the IC oscillation features and bulk parameters of ions with a nonzero initial drift velocity by solving directly the Boltzmann equation coupled with ion equations of motion in a temporally-constant electric field (note that this "constant" occurs only instantaneously for a stochastic field). To give a clear illustration for the IC oscillations in solitons' stochastic electric field, we relax the "constant" condition in Section 4 by using an artificial stochastic E which has 10-step random strengths in corresponding 10 different time intervals. The accommodation of ω (as well as other parameters) is illustrated in response to the E switches. At last, the work is generalized by using two random-number generators for the strength and time, respectively, in Section 5. In Section 6, we summarize the results and have a discussion. The last Section gives conclusions.

PHYSICAL MODELING OF NONLINEAR WAVES
In order to provide the most basic picture for the new mechanism, and thus to gain important insights into more complicated situations, while still being able to illustrate the process clearly, we set up a physical model in a cylindrical geometry (r, ϕ, z) for a dense cluster of soliton trains propagating in a magnetic flux tube with the axis along the magnetic field B = Bêz (wherê ez is the unit axial vector), which is assumed to be homogeneous in space, as described by Fig.1. The background to set up such a model is identical to what was presented in Ma & St-Maurice [60] where we focused on the auroral region which is in and above 140 km, well with the F-layer. In this region, numerous data were collected by satellites, radars and rockets which exposed the existence of solitary waves, as introduced in the above Section. Based on previous studies, this region owns a magnetic field of 0.5 Gauss; ion temperature of 1000 K; stochastic transverse electric fields up to an order of 50 mV/m or greater; an ion-neutral collision frequency of an order of 0.01-1 Hz; and, ion gyrofrequency of an order of 50 Hz. See [60,66] in details. The consideration of the modelling is inspired by FAST observations (e.g., [61,62]) which provided the features of the soliton cluster in the tube as follows [67]: (1) the cluster is composed of infinite long cylindrically-symmetric trains of solitons; (2) each of the trains appearing in space has the same space-charge density δnsc; (3) both the number and time are random for trains to emerge in space due to the plasma turbulence; (4) space charges carried by all the trains arising in space cause a homogeneous density perturbation to the uniform and isotropic background n0; (5) the cylinder-edge effects are neglected with a characteristic radius Rc of a circle (heavy dashed line) which is well inside the flux-tube cylinder (heavy solid boundary line); (6) the electron-electron, electron-ion, electronneutral, and, ion-neutral interactions are neglected, and thus we are dealing with a collision-free problem.
Under these simplifications, the space-charge density of the whole cylinder, δnc, should be δnsc if the flux tube is completely filled with soliton sets. However, soliton trains appear in space stochastically. Thus, δnc is stochastic, denoted by δñc, where and hereafter the sign "˜" means "stochastic". This stochasticity originates from the probabilistic magnitude and appearance of δnc in the flux tube. In the area within the radius r (thin solid line), δñc produces a radial electric fieldẼr by means of Gauss's law if we assume that solitons have electron space charges: δñc is positive. Clearly, the electric field within the cylinder is proportional to r, and points radially inward. We can check that the plasma is quasi-neutral even if the plasma has a density perturbation: if we were to letẼc=2 V/m at Rc=1 km, the corresponding space-charge number density would need to be δñc ≈ 2 × 10 5 m −3 . This is 10 4 to 10 6 times smaller than the ambient plasma density n0 in the ionospheric F -region. In such a radial electric field proportional to r, at any time beforeẼc switches to a new amplitude next, ion dynamics are determined by the electrostatic, spacecharge electric field (crossed to B) which is constant with time at the present step. We study the ion oscillations starting from t = 0 when the electric fieldẼr is applied, and ions have an initial state with a Maxwellian distribution function, but the initial guiding-center (GC) velocity v d0 of ions (or, the drift velocity exhibiting the initial state of the ion's instantaneous center in velocity space) is nonzero.
We are going to deal with a system of a nonrelativistic ensemble of ions of the same species in the absolute space. The particles are assumed identical, and a test particle thus provides an idealized approximation to exhibit physical properties of the specified domain [68][69][70]. We use the ion's Hamiltonian formulism as the radical basis for the study: 2Ẽ cRc · (r/Rc) 2 is the electric potential, and A is the vector potential satisfying B = ∇ × A. In cylindrical coordinates, where and hereafterẼc is written as Ec for simplicity. These two expressions give and then, in which Ω = eB/mi is the ion gyrofrequency. Using the two canonical Hamilton's equations of motion,ṙ = ∂H/∂p andṗ = −∂H/∂r, we obtain:r Because the time-inversion transformation has an unaltered nature [68], that is, for the inversion of the time direction (t → −t), there exist (r → r) and (v → −v), we know that the position vector r and hence all quantities that depend only on r do not change sign; by contrast, the velocity vector v and quantities that depend only on v change sign. This property gives that for two states {r1, v1, t1} and {r2, v2, t2} of a particle, there are two identical solutions for the same equation of motion. One provides expressions of {r1, v1, t1} by using {r2, v2, t2}, and the other is to express {r2, v2, t2} by {r1, v1, t1}, whereas the description of the characteristics of motion is unaltered. That is to say, we can either use the initial state as the final state, or, vice versa. For ions, Eq.(2.5) provides three constants of motion due to the fact that the Hamiltonian does not contain ϕ, z, and time t explicitly: the azimuthal angular momentum p ϕ = K, the axial momentum pz, and the total energy H. Expressed by the parameters at the two states, we have vz1} and {r2, vr2, v ϕ2 , vz2} the parameters of {r, vr, v ϕ , vz} at t = t1 and t = t2, respectively. Thus, in the plane perpendicular to B, we obtain two Hamilton's canonical equations: corresponding to the conservation of canonical angular momentum and that of the total energy, respectively. Concisely, Eq.(2.8) gives one modified Hamilton's canonical equation of motion: Following exactly the same algebra as given in [60], we obtain the solution of Eq.(3.1) in the case when the initial GC velocity v d is nonzero, in which a00 = a0|t=0 = 1 is the initial value of a0; Rv is the distance between the GC velocity v d and the E × B drift velocity vE = E × B/B 2 ; ω is the ion cyclotron frequency to be determined later. It is clear that the parameters a0, v dr , and v dϕ are functions of t and r but irrelevant to velocity components vr and v ϕ . In concise vector form, the top two expressions in Eq.(3.2) are expressed as follows: which shows that the motion of ions contains two circular trajectories in velocity space: one is that v rotates around the GC velocity, v d , and the other is that GC velocity rotates around the E × B-drift velocity. If assuming v d0 = 0 in Eqs.(3.2,3.3), we immediately obtain results given by Eqs.(16∼20) of [60], which studied the v d0 = 0 case. Fig.3 in that paper showed the two orbits in velocity space with an origin O. As a generalization of that case for v d0 ̸ = 0, the two velocity-vector circles are now in a new frame which has a shifted origin O ′ , as presented in Fig.2. In the new frame, the GC v d -circle passes through O ′ .

Fig. 2. Characteristics of ion motion for v d -circle not passing through the origin O. A shifted frame is introduced to satisfy v d -circle passing through the new origin O ′ . The relation of the two frames is vr
In the old frame, the initial In the old frame, the initial drift velocity at t = 0 is v d0 ̸ = 0, and the E × B drift is still as before: (3.4) which does not equal to the magnitude (Ec/B)(r/Rc) of the vector vE. Thus, the origin O is either inside the v d circle, or outside it. Fig.2 shows the latter case based on the assumed initial condition: |v d0 − vE| < |vE|.
By contrast, in the new frame, we introduce a "pseudo-electric field": and then, the difference between vE and v ′ E is in the ϕ-direction: in which (3.8) Using this shift, we obtain a relation between the old and new frames: and the expression of Rv changes from the old frame to the new one: in the circular motion at any time t ≥ 0, we have in the new frame, or, using the scalar expression, which indicates that the GC velocity v ′ d passes through the origin O ′ , and there is an initial GC velocity v ′ d0 = 0 at t ′ = 0. Therefore, we have following important relations in the new frame, just as those relations in the case of v d0 = 0 at t = 0 in the old frame given in [60]: in which Eq.(3.6) is used. This expression gives the IC frequency in the case of v d0 ̸ = 0. It tells us that there are two factors which determine the ion cyclotron oscillation ω: the strength of the electric field E in which ions are residing, and the initial GC velocity of ions, v d0 . If E = 0 and v d0 = 0 at the same time, ω returns to Ω; If E ̸ = 0 but v d0 = 0, ω is given by the third expression in Eq.(14) of [60]; If E = 0 but v d0 ̸ = 0, we have Rv = |v d0 |, and ω is given by: which discloses that if ions have a bulk speed in a magnetic field, they will oscillate in a cyclotron frequency which is different from the magnetic gyrofrequency. This is an interesting feature: even though there is no electric field in space, ions can still keep cyclotron oscillations as if they were in an electric field. If E and v d0 are all nonzero in the case under discussion, we rely on Eq.(3.16) to calculate ω in a stochastic electric field produced by solitons' space charges.
We are dealing with a case of v d0 ̸ = 0 in the old frame: v d0 = {v dr0 , v dϕ0 } ̸ = 0. Viewed in the new frame, this initial condition is, by using Eq.(3.9): which means that ions start to oscillate from an initial time t ′ = t1 in the new frame with a nonzero GC velocity vector v ′ d0 | (t ′ =t 1 ) ̸ = 0 which was evolved from v ′ d0 | (t ′ =0) = 0 in the new frame. The initial time t ′ = t1 is determined by a relation obtained after applying the initial condition v ′ d0 to the last two expressions in Eq.(3.15): , or Before the end of this section, we show the expressions of ion bulk kinetic energy 1 2 mi⟨v⟩ 2 (where ⟨v⟩ is the ion bulk velocity, or the average velocity) and ion temperature T (hereafter the subscript "i" in Ti is omitted for simplicity) by employing the ion velocity distribution, in order to show the effect of ion cyclotron oscillations on the evolutions of observable ion properties in the stochastic space-charge electric field of solitons.
The ion distribution function fi under collisionfree conditions is obtained from the following Boltzmann equation: in which the electric field E is "external", produced by the solitons' space charges. Macroscopically, this field is stochastic, maintained by the dynamical processes of the propagation of solitons which are unaffected by the local behavior of the ions; microscopically, within any tiny temporal intervals (t, t + ∆t) when a specific space-charge electric field appears in space with a lifetime ∆t, ions are residing in this "external" constant field from t to t + ∆t. In this sense, we are solving a Boltzmann equation the solution of which can be accessed purely analytically, according to our previous work, rather than a Boltzmann-Vlasov equation (where the electric field in the equation contains both 'external' and 'internal' components) which requires numerical calculations during which the physical mechanism of any resultant effects caused by the stochastic field is more difficult to be recognized.
As discussed in [60], the function fi describes the probability of finding a particle (exactly, "an ion" in our case) in a particular volume element drdv around the phase-space point {r, v} in the 6-dimensional phase space filled with identical particles [68]. Eq. (3.20) states that in the absence of the short-range collision term (∂fi/∂t) c , fi remains constant along the 6-dimensional trajectories followed by the ions in phase space, once a particular initial condition is stated [69,70]. To be more specific, if we know the initial ion distribution function at an initial phase-space vector point {r(t0), v(t0)} = {r0, v0} in any tiny temporal intervals, we are able to describe the ions' distribution function fi(r, v, t) at any time in the intervals in phase space, namely, the problem Dfi/Dt = 0 is simply formally given by   fi(r, v, t) = fi(r0, v0, t0) = f0(r0, vr0, v ϕ0 , t0) (3.21) Let ni represent ion density, and T = {Tr, T ϕ , T ∥ } represent the temperature where Tr, T ϕ , and T ∥ are the three components in radial, azimuthal, and axial directions, respectively. By assuming that the initial distribution f0 of ions is a time-stationary, position-independent, velocityshifted, density-homogeneous, and temperatureanisotropic Maxwellian with nonzero initial GC velocity v d0 , initial temperature Tr0 = T ϕ0 = T ∥ = T0, initial density n0, and initial time t0 = 0, we have: Here, we employ a "Backward mapping" approach to transform this initial distribution f0 to the final solution of the Boltzmann equations. As introduced in [60,66], the core of this approach lies in the task of finding an explicit connection between the initial phasespace state, {r0, v0, t0}, and the final one, {r, v, t}. Luckily, this relation can be obtained if we solve the set of differential equations of motion as done in [60]: there are two identical solutions for the same set of equations of motion; one provides {r, v, t} expressed by using {r0, v0, t0} (forward mapping), while the other provides {r0, v0, t0} expressed by using {r, v, t} (backward mapping), whereby the description of the characteristics of motion is traced backwards but is otherwise unaltered. We use the latter method to solve fi(r, v, t). This method allows us to relate the 6-dimensional phase point {r, v} at any time t to {r0, v0, t0} and to therefore find the distribution at time t, since the initial distribution is already fully known, as given by Eq. (3.22). Finding the distribution function is then just a matter of expressing r0 and v0 in terms of r, v, and t in the expression for the initial condition f0. Therefore, Eqs.(3.2,3.21) thus give from which we have ni, average velocity ⟨v⟩ = {⟨vr⟩, ⟨v ϕ ⟩}, and T as follows: Consequently, the ion bulk kinetic energy 1 2 mi⟨v⟩ 2 and temperature T are expressed by

OSCILLATIONS IN A STOCHASTIC NONLINEAR WAVEFIELD
In a period of time 1 s, we artificially choose ten random electric field strengths of Er to produce ten Ec/B values at ten random moments ti (i = 1 ∼ 10) in sequence, as shown in Column A and B, respectively, in Table 1. Column C gives (Ec/B)(r/Rc) at r = 0.5Rc. Hereafter, the subscript numbers from 1 to 10 attached to all physical parameters indicate the corresponding Er-levels, respectively.  ∈ [t1, t2]. The initial conditions are as follows: a01(t1) = a001(t1) = a −1 001 (t1) = 1, v dr1 (t1) = v dr |t<0 = 0, and v dϕ1 (t1) = v dϕ |t<0 = 0, along with δv ϕ1 = ϕ01 = 0. This is the case described in great details in [65] when the v d -circle passes through the origin. The input parameters of this Table 1. Parameters of Stochastic heating in 10 Random steps of 0 ∼ 1 s at r/R 0 = 0.5

E
Notice that the ion cyclotron oscillation frequency ω is written in its genuine form: it is as a matter of fact determined by Rv, rather than Ec/B; however, if the initial ion GC velocity v d is zero, we have Rv = Ec/B (a case discussed in [60]. In t ∈ (t1, t2), these input parameters give rise to the ion characteristics of motion at an arbitrary time t as follows: At the end of the stage, t = t2, above functions have following values, respectively: We know that a01(t2) and {v dr1 (t2), v dϕ1 (t2)} represent the ion temperature and GC velocity at t = t2, respectively. They keep invariant during the switch of the electric field from level 1 to 2:

E r -level 2 in t ∈ [t 2 , t 3 ]: Oscillations with ω 2 for generalization
Without loss of generality, we give expressions for the ion oscillation features at this level after the electric field switches from Ec1/B to Ec2/B. By using the initial conditions given by Eq. (4.4), we obtain the input parameters as follows: ] a 02 (t 2 ) (4.5) which gives rise to the ion characteristics of motion at an arbitrary time t as follows: At the end of the stage, t = t3, above functions have following values, respectively: (4.7) which keep invariant during the switch of the electric field from level 2 to 3, and behave as the initial conditions for the new step:
Secondly, though solitons' electric field stimulates ions from the gyration frequency Ω to the stochastic cyclotron frequency ω, ω never returns to Ω even if the electric field is turned off. Look at Er-level 5 and 10 in the upper left panel of Fig.3, where the electric field strengths are zero. However, the ω/Ω-values in the lower left panel are 1.986 and 2.548, respectively, not 1 (or, ω = Ω) as normally considered it should be in the absence of electric fields. This indicates that when the electric field disappears, ions still oscillate as if there were an electric field. As discussed in the last section, this "imaginary" field is nothing but as a matter of fact a reflection of the nonzero drift velocity that ions have acquired before the electric field is off.  Table 1. Thirdly, during the stochastic cyclotron oscillations, ions are heated transversely, as revealed by the evolution of a −1 0 (t) in Fig.4. As discussed in the last section, a −1 0 (t) reflects the ions' transverse temperature Tr (or T ϕ ) evolving from the initial isothermal T0 in the soliton's space-charge electric field proportional to the radius. Fig.4 shows that a −1 0 (t) is mostly higher than 1. This means that T is often above T0. The figure shows that ions are heated to 1.36 T0 on average, changing from 0.3T0 to 5.8T0, while Table 1 tells us that ω varies from 1.915Ω to 3.093Ω during this period of time. Thus, cyclotron oscillations and transverse ion heating are the both available manifestations of the presence of the solitons, or, exactly, their space-charge electric field in space at any time. It is worth to mention here that the last two columns a00 and a −1 00 in Table 1 are retrospective values of a0(t) and a −1 0 (t), respectively, at t ′ i = 0 in the shifted frame with the origin O ′ .
Lastly, in sharp contrast to the random amplitudes of v dr (ti) and v dϕ (ti) caused by the electric field switches, as shown in the upper right panel of Fig.3, the GC-velocity (or, the v d -vector) evolves in circles with different radii in velocity space during these switches, as shown in Fig.5. All the v d -circles are symmetric to the axis of v dr (t)=0. This means that the mean amplitude of v dr (t) is always zero. Thus, v d rotates in ϕdirection on average with a temporally changing speed, as shown by, e.g., Eq.(4.6). (Ec/B)(r/Rc), Rv, δv ϕ , and ω/Ω

OSCILLATIONS IN A GENERALIZED STOCHA-STIC NONLINEAR WAVE-FIELD
In order to get deeper insights into the features of the ion cyclotron oscillations triggered by stochastic space-charge electric fields of solitons, we use two random-number generators to simulate the stochastic appearances and the Ec/B strengths of electron solitons. The maximum temporal interval is 1 (in unit of one gyro-period 2π/Ω s). The lifespan of the stochastic electric field is 4000 gyro-periods. The calculation also gives results for extra 1000 gyroperiods to see what happens after the electric field is switched off. We consider ions at r = 0.5Rc.
For comparisons, we consider both a lowamplitude stochastic electric field of solitons and a high-amplitude one, corresponding to two extreme cases FAST could encountered. The peak Ec/B-strength in the former is 4 (in unit of v th = 1 km/s), while RcΩ = 3 (in unit of v th ); by contrast, the peak Ec/B-strength in the latter is 90, while RcΩ = 275. The two corresponding nominal ion gyration frequencies (that for Maxwellian ions with zero GC velocity to response to the peak stochastic electric field strength) are 2.52Ω and 1.52Ω, respectively.  in this case. The upper panel shows that that the kinetic energy is enhanced by at least 2 orders of magnitude by the stochastic electric field. For the ion temperature, it also demonstrate markedly up to a 20-fold increase. Because in the present study the the parallel component is assumed constant, the temperature growth is contributed by the transverse components Tr and T ϕ . After 4000 gyrations of time, both v 2 d and T keep constant in time, much higher than their respective initial values. Fig. 8 and Fig. 9 manifest the four parameters and v 2 d & T , respectively, in the high-amplitude stochastic field. From the upper left panel of Fig.  8, we see that (Ec/B)(r/Rc) is much higher than that in Fig. 6, which produces much higher Rv, δv ϕ . On the contrary, the cyclotron frequency ω in the lower right panel of Fig. 8 is lower than that in Fig. 6. By checking analytical expressions in the last section, we know that ω/Ω is really related to the electric field (Ec/B), but depends on the ratio between (Ec/B) and RcΩ. Though (Ec/B) increases 22.5 times the weak field, the fact that RcΩ becomes 91 times the weak field makes the ratio is smaller than that in the weak field. Thus, ω has smaller amplitudes which are within 1 ∼ 7Ω in frequencies. The average is around ∼ 4.4Ω which seems to be the final value stabilized after 4000 gyro-periods of time. Strikingly, the gained kinetic energy of ions is much higher in Fig. 9 than that in Fig. 7, while the ion temperature shown in Fig.9 does not increase that much as that in Fig. 7. For the two cases, the IC oscillation frequency shifts around ∼ 5Ω on average, between 4.4Ω and 5.5Ω. This indicate that observable frequencies should not fall exactly about Ω. Instead, they should be away from the ion gyro-frequency. In the present radially inward electric field case, ω has a "blue" shift to the LH frequency. Reasonably, if the electric field is radially outward (say, produced by positive space-charge solitons), we predict ω will have a "red" shift to a frequency lower than Ω. Another paper will discuss this case.

SUMMARY AND DIS-CUSSION
Plasma instabilities and waves in IC modes is a hot topic in traditional linear wave theories, where the electric field is weak, considered as a perturbation under a plane wave formulation. Aiming at illustrating the IC oscillations in response to a stochastic, strong electric field contributed by nonlinear waves in auroral ionosphere, we established a physical model to describe characteristics of ions residing in such an electric field which randomly appears in space with random strengths, however, is instantaneously constant in time. To get easy access and comprehensive insight into the oscillation features, we relied on ion Hamiltonian mechanics to solve the Boltzmann equation completely analytically for an invariant field with time. We obtained a double-circle trajectory in velocity space for ions with a nonzero initial GC velocity, and acquired evolutions of their kinetic energy and temperature. Then, we designed a stochastic electric field with 10-step strengths to illustrate how the IC oscillation frequency and other parameters accommodate to the strong space-charge electric field of solitons which appears in flux tubes randomly with time. After that, we bring two random-number generators into play to simulate real situations at a position well inside the flux tube. We obtained that the IC frequency can be shifted to several ion gyration frequencies. Specifically, with both a low-amplitude electric field and a high-amplitude one, we reported that the actual IC frequencies fall in around 5Ω on average of the two cases.
The simulation result coincides with observations, e.g., Fig.1(aa-cc) in [61]. The data gives five pulses in ∼ 5 ms, indicating a frequency of ∼1 kHz for solitary structures to appear at a specific spatial position. Using H + gyrofrequency Ω ∼200 Hz we calculated in Section 1 from FAST measurements, we predicted from our result that solitons should have a frequency of 880∼1100 Hz, in agreement with what FAST data exhibited. Here, we would like to show our special concern to the authors' claim that the frequency of the evenly spaced solitons appears to be, "within error", the LH oscillations (see Fig.5 in that paper). In the linear wave regime, the LH instabilities are triggered by the density and/or temperature gradients of the background plasma, involving both modulations of the perturbed electric field to electrons and ions. In upward/downward current regions FAST satellite passes, such conditions to excite LH waves seem not satisfied in the linear wave regime. If LH oscillations do exist, they may be triggered by some kind of mechanisms related to strong soliton electric fields. Unfortunately, the authors mentioned nothing about the activation of the LH instabilities. We are going to pay attention to this issue and show linked evidence to support the authors' argument.
This paper is the first attempt to account for measurements of plasma IC oscillations by employing a nonlinear plasma wave model. The approach hires the kinetic theory, instead of the linear wave theory, to understand the IC oscillations observed in space, where the linear plasma process has evolved to produce measurable nonlinear phenomena by highresolution payloads. As a start, we attacked the most basic problem with as simple a model as possible toward our goal to gain important insights into more complicated situations, while still being able to obtain applicable numerical solutions to observations. In the model, we did not take into consideration the effect of the parallel electric field sustained by solitons on perpendicular ion oscillations, and neglected an upward mirror force which may be brought about by a diverging magnetic flux tube. We exclude the boundary effects on the characteristics of ions by simply choosing a radius which is well inside the flux tube cylinder. We also assumed that the solitons' space charges are distributed in the flux tube uniformly, regardless of a possible density inhomogeneity due to intermediate spaces among soliton trains. Though with these limits, the feature of measured IC oscillations driven by nonlinear waves can be satisfactorily explained. We hope the work could provide a reference to future studies on solving similar problems under conditions that traditional linear wave models are no longer suitable.

CONCLUSION
After Chiueh & Diamond predicted the existence of "space-charge Clumps, numerous highresolution observations (e.g., FAST, Polar) confirmed these space-charge structures which were nothing else but the measured nonlinear electrostatic solitary waves (see the first report in [56]). These stochastic solitary structures are space-charge carriers which contribute strong transverse electric fields in space plasmas to excite ion cyclotron (IC) oscillations. Different from the stochastic methods used in different fields of physics [63][64][65], this paper offered an alternative for readers to deal with similar studies in the elucidation of the IC oscillations stimulated in auroral ionosphere. This new approach was featured by a couple of recognitions in treating the space-charge electric field, as demonstrated in [67]: macroscopically, the field is stochastic, maintained by the dynamical processes of the propagation of solitons unaffected by the local behaviour of the ions; microscopically, within any tiny temporal intervals when a specific space-charge electric field appears in space with a lifetime ions are residing in this externally constant field. In that sense, we are solving a simpler Boltzmann equation analytically, rather than a more complicated Boltzmann-Vlasov equation which requires numerical approaches. For the stochastic, strong space-charge electric wavefield E of nonlinear waves propagating in auroral ionosphere, we conclude that (1) With a nonzero initial guiding-center (GC) velocity, ions are found to follow a doublecircle trajectory in velocity space with an IC oscillation frequency ω which shifts from the magnetic gyrofrequency Ω = eB/mi; (2) After the "constant" condition of the field is relaxed by using a simple stochastic E strengths in different limited random time intervals, frequency ω accommodates the E switches which brings about variations of related transport parameters; (3) By generalizing the stochastic properties in both the field strength and time interval, ω can be shifted to several times over the value of Ω, bringing astonishing enhancements in the physical properties such as, temperature, kinetic energy, in good agreement with what observations demonstrated in auroral fieldaligned current regions.