Simulating the Ring Current Proton Dynamics in Response to Radial Diffusion by Ultra‐Low‐Frequency (ULF) Waves

Radial diffusion (RD) induced by ULF waves can contribute to particle acceleration and scattering. Past global simulations that incorporate RD often use dipole magnetic fields, which could not realistically reveal the role of RD. To better understand the effects of RD and identify whether a background magnetic field model matters in understanding the ring current dynamics in response to RD, we simulate a storm event with different magnetic configurations using a global kinetic ring current model. Results indicate that RD can effectively diffuse protons of hundreds of keV to inner regions (L ∼ 3.5), especially in recovery phase. Comparisons with in‐situ observations demonstrate that simulations with TS05 overall capture both the intensity and variations of proton fluxes with the aid of RD, whereas that with a dipole field significantly overestimates low‐L region fluxes. This study implies adopting realistic magnetic fields is important for correctly interpreting the role of RD.


Introduction
Ultra-low-frequency (ULF) waves, with a frequency range of about 1 mHz to 1 Hz, play an extremely important role in modulating higher-frequency fluctuations, transferring mass and energy, and resonating particles in the magnetosphere and ionosphere system (Bell, 1976;Cao et al., 2013;Chen, 1974;Fei et al., 2006;Li et al., 2011;Liu et al., 2009Liu et al., , 2010;;Zong et al., 2009).ULF waves interact with particles via drift or drift-bounce resonance and diffuse particles radially.Previous studies have shown that the radial diffusion by ULF waves plays an important role in modulating the distribution of particles near the Earth.For example, ULF waves can influence the distribution of high-energy protons (tens to hundreds keV) and electrons (tens of keV to MeV) and trap particles based on the mirror effect (Liu, Zong, et al., 2016), and can also lead to excitation of other waves, such as chorus waves (Li et al., 2023).Different models of radial diffusion coefficient D LL based on observations from different satellite missions have been developed (Ali et al., 2015;Brautigam & Albert, 2000;Brautigam et al., 2005;Liu, Tu, et al., 2016;Ozeke et al., 2012;Yan et al., 2023) to quantify the influence of radial diffusion.
Ring current, as a primary contributor to the Dst index, is of great significance for studying the evolution of magnetic storms (Berko et al., 1975;Daglis et al., 1993;Sugiura, 1963;Williams, 1981).Many studies have analyzed the transport and acceleration of ring current particles caused by the radial diffusion mechanism.For example, Sheldon and Hamilton (1993) used the standard radial diffusion model to study ring current particle transport and their results indicated good agreement with the ion data with energy greater than 30 keV at L > 4. They demonstrated that radial diffusion is the primary transport and acceleration mechanism leading to energetic ion flux enhancement.Jordanova and Miyoshi (2005) simulated the effects of magnetospheric convection and radial diffusion on storm time injection of high-energy ring current particles.They found that magnetospheric convection is the primary injection mechanism, while the radial diffusion mainly moderates high-energy particles.Moreover, Gkioulidou et al. (2016) analyzed the development of ring current protons using Van Allen Probes data and concluded that for low-energy protons (<80 keV), convection is the primary mechanism of transport and acceleration, while for high-energy protons (>100 keV), radial diffusion is more dominant in particle acceleration.Recently, based on a one-dimensional radial diffusion model including the charge exchange mechanism, Lyu and Tu (2022) simulated the transport and acceleration of ring current ions at different L shells, suggesting that radial diffusion is the dominant mechanism for >75 keV ring current protons at L * = 3.5-5.5.
Previous modeling studies on the effect of radial diffusion are carried out simply using a dipole magnetic field (e.g., Jordanova & Miyoshi, 2005) or using 1D models (e.g., Lyu & Tu, 2022).The role of radial diffusion on the global evolution of ring current particle distribution under more realistic magnetic field conditions needs to be further studied.Therefore, this study employs a global kinetic ring current model implemented with the TS05 magnetic field configuration (Tsyganenko & Sitnov, 2005) to analyze the responses of energetic protons to radial diffusion.Results are further compared to those under a dipole magnetic field as well as in-situ satellite observations.We found that the role of radial diffusion is more prominent during recovery phase, that using a dipole field could significantly overestimate the effects of radial diffusion in low-L regions, and that using TS05 can reasonably reproduce the observations aided by the radial diffusion.

Model Desciption
In this study, we use a global kinetic ring current model that takes into account all the radial, azimuthal, energy, and pitch angle dependences for the ring current distribution functions.As the time scales of the ring current transport and acceleration processes are much longer than that of the particles' gyration and bounce motion, we use the following bounce-averaged Fokker-Planck equation to resolve ring current particle distribution function Q, following Jordanova et al. (1997Jordanova et al. ( , 2006Jordanova et al. ( , 2010Jordanova et al. ( , 2016)): where Q(L, ϕ, E, μ, t) is the distribution function in the magnetic equator, p is the relativistic momentum of the particle, L represents the radial distance in the equatorial plane normalized by the Earth radius Re, ranging from 2.0 to 6.5, ϕ is the longitude, E is the kinetic energy from 0.15 to 500 keV, and μ 0 is cosine of the pitch angle from 0 to 90°.The brackets represent the bounce average, and h is defined as the normalized half-bounce path length (see Jordanova et al. (1996)).
The second to fifth terms on the left hand side of the equation describe the variation of distribution function due to particle convections in radial, azimuthal, energy and pitch angle directions.The bounce-averaged radial and azimuthal drift velocities, < dL/dt > and < dϕ/dt > are obtained by averaging the magnetic and electric drifts along field lines.The magnetic gradient and curvature drift at any location is calculated as: where m t is the mass for species t, and v ‖ and v ⊥ represent respectively parallel and vertical velocities.The electric drift is a combined result of convection and (3) where U represents electric potential.The convection potential is obtained from the Weimer electric potential model (Weimer, 2001).The bounce-averaged rate of change of the particle's energy and the cosine of its equatorial pitch angle, < dE/dt >, < dμ 0 /dt >, are computed following the Equations 9 and 10 in Ilie et al. ( 2012) Geophysical Research Letters 10.1029/2023GL107326 under an arbitrary magnetic field configuration.We extend < dE/dt > formula to include the relativistic scenario with the relativistic factor γ = 1 + E ̸ (m 0 c 2 ), where c is the speed of light.In this study, the TS05 magnetic field model (Tsyganenko & Sitnov, 2005) is used.
Other processes that induce changes in the particle distributions include charge exchange process between ring current ions and exospheric hydrogens, Coulomb collisions with thermal plasma, and loss cone precipitation at low altitudes into the atmosphere.The effect of ULF waves is also included as a radial diffusion term, solved in L * coordinates: where L * is inversely proportional to the third adiabatic invariant (Roederer, 2012).In this study, to avoid complex calculation of the L * , we obtain the L * parameter based on the neural network LANL * model trained by Yu et al. (2012) under the TS05 field condition.D LL is the radial diffusion coefficient adopted from the empirical formula of Liu, Tu, et al. (2016) considering only electric diffusion as the dominant diffusive process: where the unit of μ is MeV/G and the diffusion coefficient D LL is in the unit of day 1 .Note that the radial diffusion process is resolved in the (μ, K, L * ) coordinates while other previously described processes are in (L, ϕ, E, μ) coordinates.To achieve the conversion between the two coordinate systems, we calculate μ, K, and L * parameters at a given L and magnetic local time (MLT) for particles at certain energy and pitch angle, and then interpolate them into the μ, K, and L * grids to yield Q (μ, K, L * ).The new coordinate ranges are μ: 1-600 MeV/G, K: 0-0.5 G 1/2 R E , L * : 1.5-6.5.As the radial diffusion influences mainly particles with large pitch angles, we choose small K values.After solving the radial diffusion process, Q is converted back to the original coordinates for other processes.
In this study, the initial conditions are obtained statistically by quiet time data from Van Allen Probes (RBSP) A and B satellites around the magnetic equator.The boundary conditions are assumed to be Maxwellian distribution, and the characteristic temperature and density are taken from the statistical results of Kistler and Mouikis (2016).

Simulation Results
With the above model, we simulate a strong storm event that occurred on 8 September 2017 to study the effect of ULF radial diffusion on ring current proton dynamics and its dependence on the background magnetic field configuration.During the storm, the Dst index dropped rapidly to its minimum at 115 nT around 14:00 UT before entering the recovery phase.Figure 1a shows the observed and simulated SYM-H indices.Simulations yield the SYM-H index based on DPS relation (Dessler & Parker, 1959;Sckopke, 1966) after obtaining the ring current energy content from distribution functions.It indicates that the simulated SYM-H indice agrees relatively well with the observation, providing good fidelity for the following analysis.

Effects of the Radial Diffusion
We analyze the effects of radial diffusion under TS05 magnetic field conditions.We select 13:00 UT (main phase) and 20:00 UT (recovery phase) to compare the flux distribution of 180 keV protons at 80°pitch angle obtained from both simulations with and without the effect of ULF wave radial diffusion, as shown in Figures 1b-1d and 1e-1g.The normalized difference is displayed in Figures 1d and 1g to demonstrate the effect of radial diffusion at the two selected times.All other proton drift and loss processes remain the same in both simulations.As shown in Figure 1b, during storm time at 13:00 UT, protons penetrate to low-L regions (L ∼ 4) in the afternoon sector, and flux peaks around L = 4.This indicates that the transport of protons during the storm main phase is dominantly controlled by the Earthward electric convection and westward gradient-curvature magnetic drift, resulting in an asymmetric ring current.After adding the role of radial diffusion due to ULF waves, the difference in the proton distribution is shown in Figure 1d.Blue/red represent flux decay/enhancement, respectively, owing to the radial diffusion process.It can be seen that the proton flux is enhanced in the midnight-to-noon sector outside L = 4, except for a small region around the afternoon outside L = 5 where the flux is reduced.Flux reduction also appears in the dawn sector in the inner region.These changes of proton flux in the equatorial plane due to the inclusion of radial diffusion are quite dynamic in the storm main phase, perhaps a combined result of both convection and radial diffusion.In the recovery phase at 20:00 UT, the ring current gradually decays to become more symmetric.It is found that the effect of radial diffusion is more consistent across all MLTs, with the proton flux significantly increased in large L regions.
Besides the spatial distribution, the temporal evolution of the energetic (80 keV < E < 300 keV) proton flux is further examined.We choose 80 and 180 keV protons as examples to demonstrate energy-dependent responses to the radial diffusion.Figure 2a demonstrates the simulated fluxes at MLT = 6 (18) for both energies without including the radial diffusion and Figure 2b represents that including the role of radial diffusion.Consistent with the results in Figure 1, the transport of ring current protons is dominated by convection during the storm main phase.The protons with medium-energy (80 keV) on the duskside (MLT = 18) penetrate to L ∼ 3.0 after 12:00 UT in the main phase, while those on the dawnside (MLT = 6) do not approach L ∼ 3.5 until ∼15:00 UT because it takes time for them to convect westward to the dawn sector.Note that proton flux enhances intermittently near MLT = 6 during the main phase and recovery phase.Based on Movie S1 (without radial diffusion), this can be attributed to the intermittently injected plasma from the outer boundary into the inner magnetosphere through the night-to-dusk sector, followed by westward drifts toward the dawnside.After considering the influence of radial diffusion (see Figure 2b) the proton flux is slightly changed during the storm main, such as small flux enhancement outside L = 5 at dusk and near the boundary at dawn.In contrast, the proton flux is notably affected in the recovery phase.That is, the flux is increased significantly in large L regions for different MLTs as well as energies.For 180 keV, the flux enhancement also takes place in inner regions.
As the flux variations are controlled by both non-adiabatic and adiabatic processes, we therefore investigate the non-adiabatic effect of radial diffusion using phase space density (PSD).Figures 2c and 2d demonstrates the proton PSDs at μ = 40 MeV/G and μ = 80 MeV/G with K = 0.01 G 1/2 R E on the dawnside and duskside.These two μ values correspond to proton energies of ∼80 and 180 keV at L * = 4.0 respectively.The existence of vacancies in large L * areas can be attributed to the fact that the outermost L * corresponding to the L boundary at 6.5 is usually smaller than 6.5 and it also varies with time in response to time-varying magnetic field configurations.PSD outside the L * boundary is not resolved.In the simulation without considering the radial diffusion, we can see that proton PSD at μ = 40 MeV/G generally exhibits high density near the boundary, and proton PSD at μ = 80 MeV/ G shows localized peaks in low L * regions (around L * ∼ 4 at dusk) induced by the convection of 180 keV protons (see Movie S1), suggesting radial gradient versus L * (see Figure 2c).The radial gradients would enable inward or outward diffusion if the radial diffusion process is included.Indeed, after taking into account the role of ULF wave radial diffusion (see Figure 2d), significant inward diffusion is observed for μ = 40 MeV/G protons, replenishing lower L * regions.Such change is relatively clearer at MLT = 18 due to more sufficient plasma sources with large radial gradients on the duskside than that on the dawnside.The radial diffusion process also allows the localized high PSD at 80 MeV/G in small L * regions (e.g., L * ∼ 4.0) to diffuse radially outward and inward, enhancing PSD over a wide region.Clearly, during the recovery phase, radial diffusion related to ULF waves plays a dominant role in diffusing particles across a wider L shell, redistributing the plasma on a global scale.

Dependence on the Magnetic Field Configuration
One important factor associated with the radial diffusion process is its dependence on the background magnetic field.This can be easily inferred from the L * dependence of the radial diffusion coefficient, shown in Equation 5, a parameter of magnetic field geometry.Previous global simulations of ring current dynamics employed a dipole magnetic field to study the role of radial diffusion (e.g., Jordanova & Miyoshi, 2005).As L * is essentially equal to L, the computational treatment of the diffusion and convection processes under different coordinates in their studies is relatively easy.However, in this study with a TS05 model, the conversion between adiabatic coordinates and ordinary coordinates is not trivial, which involves complex grid refinement for high pitch angles and interpolation with high accuracy due to asymmetric magnetic field configurations and MLT dependence of the computed L * .These differences in the L * could inevitably influence results.
To find out whether a magnetic field model can impact the interpretation of the role of radial diffusion in simulations, we therefore simulate the same event using a dipole magnetic field configuration for comparison.Figure 3 demonstrates the normalized difference of proton fluxes caused by the radial diffusion under two different magnetic field configurations, that is, the TS05 magnetic field model (a-b) and a dipole field model (c-d).Under the TS05 configuration, the proton flux during storm main phase shows intermittent decay in large L regions, regardless of energy and MLT sectors.However, the flux is constantly enhanced in the recovery phase, with the change of the high energy flux being more prominent over a wider L range.In contrast, under the dipole case, the flux change due to radial diffusion is greatly different from the case in TS05.For medium energy at 80 keV, the flux is not affected much (light red/blue) in most MLT sectors except in the dawn sector.For higher energy at 180 keV, the flux is remarkably enhanced in all MLTs for low L regions, even in L = 3, sustained from storm main phase to the recovery phase.Only the flux near high L regions (L ∼ 5) sometimes shows decay.The low-L flux enhancement is more significant than in the case using the TS05 magnetic field model, suggesting that the transport and acceleration by radial diffusion are more influential in low L regions if a dipole configuration is used.
Which simulation provides more realistic modeling results can be assessed via comparisons with in-situ observations.Figure 4a   identify the necessity of using a more realistic magnetic field model.The green line represents satellite observations of proton flux at 180 keV with 80°pitch angle.The blue lines represent model results under dipole configuration with (solid) and without (dashed) taking the radial diffusion process into account.The red lines are model results using the TS05 magnetic configuration.Comparisons are carried out during the storm main and recovery phase from 10:00 to 23:00 UT.In general, the simulated flux with TS05 (red lines) is higher than that with a dipole field (blue lines).The former is much closer to the data in large L regions during both storm main phase (from 12:00 to 17:00 UT) and recovery phase (after 21:00 UT), while the latter is about one to two orders of magnitude lower than the data.After considering the radial diffusion, the flux is overall enhanced in the former case by about one order of magnitude particularly in large L regions, but the flux is not changed much in the latter case except near the perigee from 17:00 to 21:00 UT at low L regions, in which the flux increases significantly, overestimating the data by one order of magnitude.
We also calculate the correlation coefficient (CC) and root-mean-square error (RMSE) to quantitatively evaluate the agreement between the modeled flux and observations.While the CC in both simulations is similar, the RMSE in the simulation using TS05 is much better.In both cases, after including the effects of radial diffusion, the change in CC is not much but RMSE is improved.Results indicate that the inclusion of the radial diffusion process generally helps to bring the simulation results closer to the observation.The above comparison is done for one energy level at ∼180 keV.In fact, the results are consistent across a broad energy spectrum for hundreds of keV protons (not shown).That is, the simulation with a dipole field underestimates the energetic proton flux in large L regions, but it overestimates the low-L region flux.Overall, the inclusion of the radial diffusion under the TS05 field configuration helps to achieve a better data-model comparison.
Our results indicate that the influence of ULF wave radial diffusion appears to be different on ring current proton dynamics if we apply different magnetospheric field models (see Figure 3).After the consideration of radial diffusion, the simulation with a dipole field sees flux enhancement in low-L regions at hundreds of keV, while the simulation with TS05 sees flux enhancement in high-L regions.Such different impacts of radial diffusion could be explained by the following three aspects.First, the magnetic field configuration determines the magnitude of local magnetic fields and L * , a key parameter in calculating the radial diffusion coefficient (see Equation 5). Figure 4b shows the calculated L * for protons initially located at L = 3.5, 4.5, 5.0, 5.5, and 6.5 at MLT = 18 under the two magnetic field conditions (Dot lines represent the L * in the TS05 field while dashed lines are in the dipole field).The sudden discontinuity of L * in the TS05 field occurring at 13:00 UT is attributed to a rapid change in the global magnetic configuration, as illustrated in Figure S1a in Supporting Information S1.This is further due to a sudden change in the interplanetary magnetic field (IMF) By and solar wind pressure conditions as shown in Figure S1b in Supporting Information S1.We can see that the L * obtained in the dipole field is persistently larger than that in the TS05 field over the entire storm event, which can result in a larger radial diffusion coefficient as calculated by Equation 5.This can explain the overly enhanced proton flux in lower L regions under the dipole field condition (see Figure 3d), as a larger coefficient means stronger radially inward diffusion and thus more plasma source to small L regions.Second, for high L regions, the Δ L * within a fixed L range, indicated by colored arrows (e.g., between L = 5.5 and 6.5), is much smaller in the case with TS05 field than that using a dipole field.If we assume the same radial profile of proton distribution across L shells at MLT = 18, the PSD gradient with respect to L * is larger under TS05 conditions, which more likely allows for radial diffusion in this outer region.Lastly, we also notice that in the simulation with TS05, the L * value in the outer region (e.g., L = 6.5) is surprisingly smaller than that in the inner region (e.g., L = 5.5 or 5.0) during the storm main phase from 13:00 to 16:00 UT.The abnormal distribution of L * may be caused by excessively stretched local magnetic field configuration, as illustrated in Figures S1c and S1d in Supporting Information S1.The pronounced stretching of the local magnetic field line (or local magnetic dip) at L ∼ 6.0 results in abnormal distributions of the magnetic field line footpoint traced from the same local time but different L locations as shown in Figure S1e in Supporting Information S1.The above reversed spatial distribution of L * is therefore associated with a locally distorted magnetic field configuration, which is more stretched than that inside or outside.If the PSD is higher near the L * boundary, as shown for both energies in Figure 2, then the PSD is diffused inward to lower L * regions, in other words, to higher L regions.

Conclusion
ULF waves in the magnetosphere are believed to contribute to the particle scattering and subsequent acceleration or precipitation down to the upper atmosphere.We studied the ring current proton dynamics considering the ULF wave-induced radial diffusion during a storm event.Two different magnetic field configurations were employed in simulations to identify their impact on interpreting the role of radial diffusion.We found that in the simulation with TS05, energetic (80 keV < E < 300 keV) ring current protons can be diffused into L ∼ 3.5 by ULF waves, and increase the flux in L > 3.5 regions, especially in the recovery phase.Comparisons with in-situ observations indicate that the high-L region flux is reasonably captured by the model after the radial diffusion is taken into account.However, the simulation using a dipole field showed an overly enhanced flux in the inner region (L < 4) after including the radial diffusion, significantly overestimating the observations.On the other hand, the high-L region flux is considerably underestimated even with the radial diffusion.
The behavior of the flux variation due to the radial diffusion is quite different in the two simulations, because the radial diffusion highly depends on D LL and the PSD gradient versus L * , both of which are L * related and hence magnetic field geometry dependence.This study therefore demonstrated the significance of adopting a more realistic magnetic field model, even in the inner magnetosphere, for correctly understanding the role of waveparticle interactions.

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Radial diffusions are able to effectively diffuse energetic (80 ∼ 300 keV) ring current protons to L ∼ 3.5 especially during recovery phase • Simulations with a dipole field may overestimate the role of radial diffusion in low L regions, but underestimate in high L regions • Adopting a more realistic magnetic field model is necessary to correctly interpret the role of radial diffusion Supporting Information: Supporting Information may be found in the online version of this article.

Figure 1 .
Figure 1.(a) The simulated SYM-H index with ULF waves (orange) and the observed SYM-H index (black) during the storm event that occurred on 8 September 2017.(b, c) The flux distribution of 180 keV protons with 80°pitch angle without (b) and with (c) the radial diffusion associated with ULF waves at 13:00 UT.(d) The normalized difference between (b, c), defined as flux [with] flux [without] ) ̸ flux [without] .Blue/red means flux decay/enhancement.(e, f) The flux distribution of 180 keV protons with 80°pitch angle without (e) and with (f) the radial diffusion at 20:00 UT. (g): the normalized difference between (e, f).

Figure 2 .
Figure 2. (a, b) The temporal evolution of 80 and 180 keV proton flux at MLT = 6 and MLT = 18 without (a) and with (b) the radial diffusion, respectively.(c, d) The variation of proton PSDs at μ = 40 and 80 MeV/G with K = 0.01 G 1/2 R E at MLT = 6 and MLT = 18 without (c) and with (d) the radial diffusion.
compares the model results to proton flux measured by the Radiation Belt Storm Probes Ion Composition Experiment (RBSPICE) instrument onboard RBSP-A to assess the model performance as well as to

Figure 4 .
Figure 4. (a) Compares the model results to proton flux at 180 keV with 80°data (green) measured by RBSPICE instrument onboard RBSP-A from 10:00 to 23:00 UT.The blue/red lines represent model results under dipole/TS05 configuration with (solid) and without (dashed) taking the radial diffusion process into account.(b) Shows the evolution of L* of protons initially located at different L shells at MLT = 18.