In the study of space plasma, it is advantageous to describe each constituent species utilizing a distinct velocity distribution function, denoted as fs(vs,rs,t). The velocity distribution function is defined such that the expression fs(vs,rs,t)dvsdrs quantifies the number of particles of species s that, at time t, possess velocities within the interval vs and vs+dvs and positions between rs and rs+drs. The temporal evolution of the species' velocity distribution function is governed by the cumulative effects of collisions and interactions, as well as the dynamical movements of the species in phase space under the influence of external forces (Schunk, 1977). This evolution can be mathematically represented by the well-established Boltzmann equation: 1∂fs∂t+vs⋅∇fs+esmsE+vs×Bc⋅∇vsfs=δfsδt. In this equation, the left-hand side represents the evolution of the velocity distribution function fs(vs,rs,t) under the effects of external forces, and the right-hand side represents the Boltzmann collision integral; here, it represents the rate at which fs changed as a result of wave particle interactions in the region of study. In the above equation, E is the polarization electric field; B is the geomagnetic field; c is the speed of light; ∇ is the coordinate space gradient; ∇vs is the velocity space gradient; and es and ms are the charge and mass of species s, respectively. The suitable expression for (δfs/δt) in the case of wave particle interactions is given by Retterer et al. (1987); they considered the effects of WPI as particle diffusion in the velocity space. 2δfδtWPI=1v⊥∂∂v⊥D⊥v⊥∂f∂v⊥ In the above, D⊥ is provided by Retterer et al. (1987) and represents the quasi-linear velocity diffusion rate perpendicular to the geomagnetic field: 3D⊥=ηq24m2|Ex(ω=Ω)|2, where |Ex(ω)|2 is the measured spectral density of the electromagnetic turbulence, η is the proportion of the measured spectral density of the plasma wave instrument (PWI) on board the Dynamic Explorer 1 (DE-1) spacecraft that corresponds to the left-hand polarized wave, q is the ion's charge, m is the ion's mass, Ω is the ion's gyrofrequency, and ω is the angular frequency of the electromagnetic turbulence.

The expression of the velocity diffusion rate D⊥ as given in Eq. () is independent of velocity and depends on position (altitude) via changes in the ion gyrofrequency Ω along the geomagnetic-field lines. By examining experimental data of the electric-field spectral density obtained by the plasma wave instrument (PWI) on board the DE-1 satellite (i.e., for high-solar-activity conditions), Barghouthi (1997) and Barghouthi et al. (1998) calculated the altitude dependence of D⊥. They came up with the following expressions for the velocity diffusion coefficient D⊥ in the polar-wind region (Barghouthi et al., 1998): 4D⊥(r)=5.77×103rRE7.95cm2s-3,for H+9.55×102rRE13.3cm2s-3,for O+. In the auroral region, D⊥(r) is given by Barghouthi (1997) as follows: 5D⊥(r)=4.45×107rRE7.95cm2s-3,for H+6.94×105rRE13.3cm2s-3,for O+. In the central-polar-cap (CPC) and cusp regions, D⊥(r) is given by Nilsson et al. (2013) as follows:

for the central-polar-cap region, 6D⊥(r)=20rRE9.77cm2s-3,for H+0.5×105rRE5.5cm2s-3,for O+, and for the cusp region, 7D⊥(r)=1.01×106rRE5.61cm2s-3,for H+2.5×104rRE6.4cm2s-3,for O+. The diffusion coefficient was given a new form by Barghouthi (2008), who discovered that it is velocity-dependent in addition to being altitude-dependent. 8D⊥(r,v⊥)=D⊥(r)1fork⊥v⊥Ωi<1k⊥v⊥Ωi-3fork⊥v⊥Ωi≥1 In the above, D⊥(r,v⊥) is the quasi-linear velocity diffusion rate perpendicular to the geomagnetic-field lines (altitude- and velocity-dependent), Ωi is the ion gyrofrequency, and k⊥ is perpendicular wave number and is related to the characteristic perpendicular wavelength of the electromagnetic turbulence λ⊥. Equation () indicates the diffusion coefficient that is dependent on altitude and velocity. However, as ions are heated and move to higher altitudes, their gyroradius may approach the perpendicular wavelength of the electromagnetic turbulence, and when the ratio (k⊥v⊥/Ωi) exceeds 1, the heating rate becomes self-limited. Bouhram et al. (2004) derived an alternative form for the altitude- and velocity-dependent diffusion coefficient and interpreted their results in terms of finite wavelength effects.

By solving the Boltzmann equation – Eq. () – using the Monte Carlo technique, the velocity distribution functions were obtained for each species (in this study, O+ and H+ ions) and their velocity moments, i.e., density ns, drift velocity us, and parallel Ts∥ and perpendicular Ts⊥ temperatures. The moments considered here are defined as follows (Barghouthi, 1997): 9ns=∫fsdvs,10us=1ns∫vs∥fsdvs,11Ts∥=msnsk∫(vs∥-us)2fsdvs,12Ts⊥=ms2nsk∫(vs⊥)2fsdvs. These Monte Carlo results will be used to calculate the mean parallel energy, mean perpendicular energy, and total mean energy as given in the following expressions (Barghouthi, 1997): 13Ws∥=12mus2+12kTs∥,14Ws⊥=kTs⊥,15Ws=Ws∥+Ws⊥, where us, Ts∥, and Ts⊥ are given by Eqs. (–), respectively, and Ws∥ and Ws⊥ are the mean parallel and perpendicular energies, respectively. Ws is the total mean energy, s denotes the type of ion (O+ or H+), and k is the Boltzmann constant.

The Barghouthi model was developed to investigate the behavior of ion outflows, specifically for H+ and O+ ions, at high altitudes and high latitudes. The simulation results generated by this model show excellent agreement with observational data from various regions, including the auroral region (Barghouthi, 2008) and the polar-wind region (Barghouthi et al., 2011). This model accounts for multiple influential factors, including gravity, the polarization electric field, the diverging geomagnetic field, and wave–particle interactions, all of which affect H+ and O+ ion outflows at elevated altitudes and latitudes. Notably, the model highlights the significant contribution of wave–particle interactions, which are responsible for ion heating. The impact of these wave–particle interactions is characterized by the velocity diffusion coefficient D⊥(r,v⊥), which has been formulated as a function of position (r/RE) along Earth's geomagnetic-field lines and the perpendicular velocity of the injected ions (v⊥). Various functional forms of the velocity diffusion coefficient D⊥(r,v⊥) have been employed in Monte Carlo simulations to derive the density, drift velocity, parallel and perpendicular temperatures, heat fluxes, and velocity distribution functions for H+ and O+ ions at high altitudes and latitudes. In the Monte Carlo simulations of this study, the appropriate velocity diffusion coefficient corresponding to the specific study region, D⊥(r,v⊥), was utilized to ascertain the temperatures and velocity distributions of H+ and O+ ions at these elevated altitudes and latitudes.

Chang et al. (1986) presented a theoretical framework for estimating the mean perpendicular, mean parallel, and total mean energies as functions of geocentric distance. This framework incorporates the average heating rate for each ion into a set of equations that describe the motion of ions along geomagnetic-field lines, as outlined below: 16Wi∥=9mi21/3rD⊥(r)(3α+1)(6α+11)2/3,17Wi⊥=(6α+2)mi21/3rD⊥(r)(3α+1)(6α+11)2/3,18Wi=Wi∥+Wi⊥=(3α+11/2)1/3mirD⊥(r)(3α+1)2/3, where W∥ and W⊥ are the mean parallel and perpendicular energies, respectively; Wi is the total mean energy; i denotes the type of the ion (H+ or O+); and α is a fitting parameter. In this theory, the mean energy ratio W⊥/W∥ asymptotically approaches a constant value.

By employing the Monte Carlo technique (specifically the Barghouthi model) alongside mean particle theory, we have derived varying altitude profiles for the mean parallel, mean perpendicular, and total mean energies across different regions of Earth's magnetosphere – specifically, the auroral region (Fig. 1) and the polar-wind region (Fig. 2) – for H+ (panels a–c) and O+ (panels d–f) ion outflow. In both methods, we utilized an appropriate D⊥(r) as the velocity diffusion coefficient for each region. Overall, our findings in the auroral region (Fig. 1) indicate an excellent agreement between Monte Carlo simulations and estimates of mean particle theory for H+ and O+ ions. It is important to highlight that, in the mean particle theory, ion heating and energization occur solely due to wave–particle interactions, which primarily depend on the value of the velocity diffusion coefficient. In contrast, the energization process in the Monte Carlo model results from the competition between wave–particle interactions and external forces, such as gravity, the polarization electric field, and the divergence of the geomagnetic field. The close agreement between the two methods can be attributed to the dominant influence of wave–particle interactions, which prevail over the effects of external forces due to the high values of D⊥(r). In other words, wave–particle interactions are the primary drivers of the energization process for both H+ and O+ ions in this auroral region. For the polar-wind region and for λ⊥→∞, this means that the velocity diffusion coefficient D⊥(r) is altitude-dependent. The values of the velocity diffusion coefficient are less than those in the auroral region; see Eqs. () and (). According to Fig. 2, there is no agreement between Monte Carlo simulations and mean particle theory estimates; this is due to the contribution of the external forces in Barghouthi model that competes with the effect of wave particle interaction. However, in mean particle theory, the external forces are not considered, and the heating is due to wave particle interaction. We report here that mean particle theory is not suitable to be used in this region, and Monte Carlo simulations are more appropriate to use, as shown in Barghouthi et al. (2011), where they compared their Monte Carlo results with observations. Also, we have found that, when we use the diffusion coefficient that depends on altitude only, its value becomes very large as altitude increases. Therefore, the values of the particle energies obtained from Eqs. () to () turn out to be very high, but when the diffusion coefficient becomes velocity- and altitude-dependent according to Eq. () (i.e., k⊥v⊥/Ω≥1), the produced particle energies turn out to be reasonable, as shown in Fig. 3 (solid blue lines and dashed red lines).

In this section, we present only the simulation results and the estimates of mean particle theory that have corresponding observations. Barghouthi (2008) compared the Monte Carlo simulation results obtained using the Barghouthi model with the corresponding observations for H+ and O+ ion outflows in the auroral region at different altitudes in the simulation tube; he obtained excellent agreement, particularly when the typical perpendicular wavelength of the electromagnetic turbulence was 8 km. Also, he observed that there is a broad agreement between the simulation results of the polar wind for this wavelength and the corresponding observations. For these reasons, we chose to use the results of the comparison with corresponding observations when λ⊥=8km, i.e., when the velocity diffusion coefficient is altitude- and velocity-dependent. We will compare the outcomes of our Monte Carlo simulations, the estimates of mean particle theory, and available observations obtained from different published articles. Observations of O+ ions at various altitudes were obtained for parallel velocity, perpendicular temperature, and parallel temperature for both polar-wind and auroral regions from Nilsson et al. (2013), and observations for parallel velocity and perpendicular temperature for O+ ions in the central-polar-cap and cusp regions were obtained from Barghouthi et al. (2016). In this study, we adopt the terms “good” and “excellent” agreement to qualitatively describe how well our simulation and theory curves correspond to observational data. These classifications are based on a visual comparison with the observational ranges, represented by the maximum, average, and minimum values. We define agreement as excellent when the simulation and theory curves largely overlap with the average observational curves and remain within the observed range across all altitudes. Conversely, we describe the agreement as good when the curves generally fall within the envelope formed by the observational maximum and minimum values, even if they deviate partially from the average trend. This qualitative approach is commonly used in comparative studies of ionospheric outflows (e.g., Barghouthi et al., 2011, 2012, 2016), and it acknowledges the significant variability typically present in observational data.

Figure 3 displays the comparison results for the auroral (panels a and c) and polar-wind (panels b and d) regions. It contrasts the Monte Carlo method (dashed red lines) and the estimates from mean particle theory (solid blue lines) at λ⊥=8km, alongside observational data represented by maximum (dotted lines), average (dashed-dotted lines), and minimum (dashed black lines) values for O+ ions. Across all four panels, it is evident that the results from the Monte Carlo simulations and the mean particle theory estimates are closely aligned, demonstrating excellent agreement at lower altitudes. At higher altitudes, while both methods display similar qualitative behavior and maintain good agreement, we attribute this acceptable and reasonable correlation to the application of a suitable altitude- and velocity-dependent diffusion coefficient in both approaches.

Based on the comparison with observations, it is evident that the simulation results and estimates from mean particle theory fall within the observational range. This indicates that D⊥(r,v⊥) is appropriate for use in both the Monte Carlo simulation and mean particle theory. Specifically, the results for perpendicular energy W⊥ (O+) from both the Monte Carlo method and mean particle theory align well with the maximum observed values in the auroral region, and they also exhibit good agreement with the average observed values in the polar-wind region. Regarding the mean parallel energy W∥ (O+), as shown in Fig. 3c and d, the simulation results and estimates from mean particle theory are closely aligned with both the average and maximum observed values in the auroral region, and they demonstrate excellent agreement with the minimum observed values in the polar-wind region.

In the central-polar-cap (Fig. 4a) and cusp (Fig. 4b) regions, we utilized the altitude-dependent diffusion coefficient from Nilsson et al. (2013), specifically when λ⊥→∞. This approach resulted in excellent agreement at all altitudes between the Monte Carlo simulations and the estimates of mean particle theory. Furthermore, we observed a strong correspondence to observational data in both regions; the results closely match the maximum and average values in the central-polar-cap region and align well with the average values from observations in the cusp region.

As a result of these comparisons, it is crucial to select the appropriate form of the velocity diffusion coefficient for each region of interest within Earth's magnetosphere when analyzing ion energization and heating processes.

Barghouthi (2008) conducted a comparative study between Monte Carlo simulations – based on the Barghouthi model – and in situ measurements of O+ and H+ ions at selected altitudes in the auroral region. This work was later extended by Barghouthi et al. (2011) to include the polar-wind region. In both studies, the simulation results sometimes showed good agreement with observational data in terms of energy magnitudes and general trends; however, notable discrepancies were also identified. These deviations were attributed to physical processes not incorporated into the model, such as variations in ion heating or cooling mechanisms, the influence of unmodeled external forces, or other dynamic plasma phenomena. Importantly, these earlier studies did not incorporate mean particle theory, which emphasizes ion heating due to wave–particle interactions.

In the present study, both Monte Carlo simulation results and theoretical estimates based on mean particle theory were found to fall within the observational range, as shown in Figs. 3 and 4. Specifically, we noted that, in Fig. 3a and b, covering altitudes between 6 and 8 Earth radii (RE), although the simulated ion energies align in terms of magnitude with observed values, the altitude-dependent energy profiles differ significantly in terms of trend. Theoretical and simulated profiles exhibit a steeper gradient with altitude compared to the observational data. Similar behavior can be seen for Fig. 4a, where the theoretical profiles show steeper gradients than those observed, particularly at higher energies. In contrast, Fig. 4b demonstrates a stronger consistency between theoretical predictions and observations.

In our analysis, we examined both absolute energy values and the slopes of the energy profiles. The discrepancies noted in Fig. 3a and b – especially the steeper theoretical and simulated slopes – can be attributed to the use of altitude- and velocity-dependent diffusion coefficients that fully account for processes like ion self-limiting heating (Bouhram et al., 2004) and localized cooling effects. The deviations in Fig. 4a are mainly due to oversimplified assumptions regarding wave mode propagation and the absence of spatial variations in plasma conditions. By contrast, better agreement in Fig. 3c and d and Fig. 4b is achieved due to the implementation of empirically based, altitude- and velocity-dependent diffusion coefficients.

These findings underscore the importance of region-specific parameterizations and highlight the need for future improvements in the modeling framework. In particular, incorporating saturated wave–particle interactions, as well as multidimensional plasma dynamics, would enhance the model's ability to replicate the observed energy profiles more accurately.