During each solar cycle, sequences of eruptive flares are followed by coronal mass ejections and interplanetary shocks, some of which arrive near Earth. At times when the solar wind enters into Earth's magnetosphere, these solar eruptions modify the dynamic conditions in geospace and trigger space weather effects like geomagnetic storms and magnetospheric substorms . Geomagnetic storms occur when the energy transfer from the Sun to geospace intensifies, as a result of the occurrence of magnetic reconnection at the dayside magnetopause during periods when the interplanetary magnetic field (IMF) has a strong and prolonged southward component . Magnetospheric substorms are caused by the variability in the north–south orientation of the IMF and evolve as energy loading–dissipation cycles . This kind of activity brings up a configuration change in the magnetosphere, including the ionosphere, and enhances the ring current and corresponding current systems flowing on the magnetopause, along the magnetotail and in the Birkeland regions see in. The associated dynamic processes evolve in a variety of timescales, from days for geomagnetic storms and hours for magnetospheric substorms down to minutes, or even seconds, for local plasma instabilities.

Solar eruptions are characterized as geoeffective when the magnetospheric response amounts to large electromagnetic perturbations, with severe consequences for the performance of ground-based power and communication networks as well as for spacecraft and weather satellites . A major goal in space weather research is to predict the dynamic state of the geospace from measured solar wind and IMF data, so as to timely distinguish those events that are harmful. In this respect, the simulation of physical processes dominating extreme space weather conditions, such as magnetic reconnection, convective plasma transport and charged particle acceleration, is required . For numerous events, magnetospheric activity can be described by means of a few geomagnetic indices, like Kp and Dst, which can in principle be derived from solar wind and IMF values. However, these indices include systematic and/or statistical errors which limit the capability to establish consistent (causal) correlations . Therefore, more detailed, large-scale numerical solvers of the coupled solar wind–magnetosphere system may be employed; such models have advanced with the increased availability of computer resources.

There are cases where global fluid and magnetohydrodynamic (MHD) simulations reproduce the observed changes in the magnetic topology to quite good accuracy . However, their use has limitations due to missing physics for the description of non-collisional processes in a multi-species plasma. Kinetic solvers and test-particle simulations, with a description of the plasma motions in adjustable physics detail, increase the reliability at smaller scales by properly addressing effects like thermal instabilities and anomalous transport . The drawback of microscopic models for global simulations is the large demand on computer resources; to cope with this, it is customary to separately model each source playing an important role in the dynamics: the ring current, the near-Earth tail currents, the radiation belts and the magnetopause. In this frame, the origin and transport of ring current and radiation belt particles during storm time, their interaction with the tail current, the escape of high-energy particles and the dynamic connection with the substorm phases have been addressed through the analysis of observations and dedicated numerical simulations .

For the description of the ring current dynamics, the plasma current distributions at the near-Earth region have been modeled in terms of the bounce-averaged, drift-kinetic equation. described the particle drifts in the storm-time field in terms of the initial and boundary particle distributions, with the coefficients in the kinetic equation calculated from the Hamiltonian description of motion. modeled the radially diffusive plasma dynamics in self-consistence with the fields by coupling the kinetic equation with a 3-D, force-balanced magnetic equilibrium code and a MHD solver for the convection electric field. In another self-consistent treatment, employed a collisionless kinetic code together with a model for the electrostatic potential, taking into account the current closure with the ionosphere. The specific model has been coupled to the code of , where it serves as the solver for the electric field.

The method we adopt in this work is to directly follow the 3-D particle trajectories under the effect of the electric and magnetic forces during the dynamic phases of the disturbance see, for example,. An advantage of studying the individual particle motions is the physics insight gained, as well as the statistics built from ensembles of particles. In such models, the Lorentz equation of motion is solved, either in its full form or reduced in terms of the guiding-center approximation, and the driving forces are (as above) the dynamic magnetic field coming from the superposition of the Earth's terrestrial magnet with the fields generated by the magnetospheric current sources , and the electric field due to large-scale plasma convection and corotation with the Earth . An important factor, however, is the modeling of the electric field component induced by the time variation in the magnetic field. The specific field is involved in the strong acceleration of charged particles which is observed during geomagnetic disturbances; however, there are a relatively low number of test-particle-based studies which have been performed in this direction like, for example,.

It becomes apparent that the modeling of the near-Earth plasma response to geoeffective solar events is of major importance for the improvement of space weather prediction. The model requirements are a consistent description of the geomagnetic and electric fields, the computation of the Sun-driven plasma dynamics and the assessment of the numerical data for the estimation of parameters related to space weather, including benchmarks against ground-based and satellite observations. In this paper, we present results from the simulation of the electric and magnetic fields and of the energetic particles in the inner magnetosphere, focusing on the ring current buildup and decay when disturbances are occurring. The physics of our model for the forces driving the plasma dynamics are cast in a form suitable for use with 3-D test-particle codes. Provided that there are suitable simulation data, a statistical evaluation for the dynamics of the different ion types is performed over the initial conditions, and an ensemble-averaged estimation of the Dst index stemming from the ring and tail current populations is given.

The structure of the paper is as follows: in Sect. 2, the physics model for the geomagnetic and electric fields is explained, accompanied by field-line tracing and equipotential contour simulations, and, following that, we describe the main aspects of the particle-tracing model. In Sect. 3 we present the numerical results: the different types of ion motion found in the disturbed magnetosphere, the statistical analysis of the particle dynamics and the estimation of the Dst index. Finally, in the concluding section, the merits of this work are summarized, the limitations of our model are discussed and further studies are proposed.

In this section, the results from test-particle simulations are shown and analyzed. The space weather scenario under study involves the occurrence of a single magnetospheric disturbance. The growth phase of the event starts at ti=0 with a quiet magnetosphere, indexed with Kp(ti)=1, and completes after tg-ti=30 min by reaching a disturbed state with index Kp(tg)=5. Then, the relaxation phase follows immediately and completes after tr-tg=5 min, during which Kp returns to its initial value, i.e. Kp(tr)=Kp(ti). The particle starts its flight at the time stamp t0, which may be before (t0-ti<0) or after (t0-ti>0) the onset of the growth phase, interacts with the disturbance until tr and continues moving under the effect of the restored fields until the time stamp t1.

In the disturbed magnetosphere, three primary types of ion trajectories are met: (i) orbits which become trapped inside the ring current, (ii) orbits that precipitate into Earth's atmosphere, and (iii) orbits escaping tailward or by crossing the magnetopause. We have computed these types by sampling various initial conditions for the ion position and energy, and the results are shown in GSM coordinates in Figs.  and . In Fig. , we have the planar projections of the orbit of an O+ ion that eventually integrates to the ring current. The motion initiates at t0=-8 min, before the event, with initial radius r(t0)=20RE, magnetic local time (MLT) ϕ(t0)=24 h, latitude θ(t0)=25∘, pitch angle α(t0)=π/2 and kinetic energy Ek(t0)=4 keV, and is followed until t1=180 min (∼ 2.5 h after the event). In Fig. , we represent the other types of motion in 3-D space for different ion species with the same input as before except for Ek: the precipitating orbit is of an H+ ion with Ek(t0)=0.5 keV, whereas the escaping orbit is of an O+ ion with Ek(t0)=7 keV.

In Fig. , the O+ ion is launched from the plasma sheet, driven towards Earth by the disturbed electric fields, and finally gets trapped in the ring current. A careful examination of the numerical data yields that, in its course to the ring current region, the ion is considerably accelerated from the energy exchange with the electric field, whereas its pitch angle has a random behavior before the entrance to the ring current and afterwards varies periodically. In the case of the H+ ion that crashes onto Earth, the orbit of which is shown in Fig. a, the particle begins with a low initial energy and is intensely accelerated, and probably due to the relation of its pitch angle with the loss cone it ends up on the terrestrial atmosphere at t=33 min (well before t1), with a final energy as large as in the previous case. Finally, in Fig. b, the O+ ion starts with a relatively high value of energy and escapes from the inner magnetosphere, along the meridian at 22:00 MLT, before t1 (at t=40 min) with a velocity gain. The different behavior of the oxygen ions for different initial energy is an indicator of the sensitivity of the ion dynamics to the initial conditions.

Regarding the particle acceleration, in Fig.  we examine the kinetic energy and the pitch angle for the motions in Figs.  and . In Fig. a, the energy of the trapped O+ ion appears to have a gain of about 1.3 orders of magnitude. The largest part of the increase occurs during the relaxation phase (tg<t<tr), where the magnetic field exhibits a steep decrease and, consequently, the induced electric field attains large values and accelerates the ions. Another incidence of energy gain occurs a little after t=2 h well inside the ring current region. This is connected to an intense pitch angle variation, as seen in Fig. b, where α(t) is displayed, which is induced by the structure (i.e. the gradients over time and space) of the local fields at the specific time. The kinetic energy of the H+ ion that crashes onto Earth also appears to have a sizeable gain (almost 2.6 orders of magnitude) at the time of reaching the atmosphere, after nearly 45 min of flight, whereas the energy of the escaping O+ ion appears a gain of nearly 1.6 orders of magnitude at the time it crosses the magnetopause, close to the end of the relaxation phase.

For the statistical analysis of the motions, numerical data have been produced over the trajectories of the ion species relevant to each territory of the magnetosphere, in loops of different initial conditions for r, ϕ, θ, α and Ek where, each time, only one or more of these quantities were varied. In Fig.  we present results for the final kinetic energy and the pitch angle from different simulations with an ensemble of Nens=1000 O+ ions, where the initial conditions varied are r(t0) and Ek(t0). In the first computation, r(t0) took values in a loop from 2 RE to 30 RE, whereas, in the other case, Ek(t0) ranged from 0.5 to 20 keV, and all the rest of the input quantities were equal to the values already defined: t0=-8 min, ϕ(t0)=24 h, θ(t0)=25∘, and α(t0)=π/2. The general picture (also implied from the above results) is that the dependence of the particle dynamics on the initial conditions is very sensitive, which is imprinted in the wide regions where Ek(t1) and α(t1) exhibit an irregular, non-smooth variation over the values at t=t0 (see especially Fig. a). However, in all cases one identifies consecutive regions where the ions either get accelerated or remain at low energy.

In Fig. a, there is a spatial region from 14 RE to almost 17 RE where all injected ions gain significant amounts of energy, as well as one within 21 RE and 25 RE where nearly all particles do not appear to have a net energization. In Fig. b the probability of acceleration appears to be larger for O+ ions with low energy at the event start (Ek(t0)<6 keV) than for initially energetic ions (having, for example, Ek(t0)>15 keV). The specific result reveals the role of the plasma sheet as a reservoir of oxygen ions which, on the course of storms/substorms, get accelerated and enhance the ring current and references therein. Finally, in Fig. c, there are regions where α(t0) varies rapidly, suggesting rotational behavior, as well as regions where the variation is slow, denoting motions close to ballistic. Most of the former ions, as implied by the inbound direction of motion driven by the large parallel velocities of the specific pitch angles, are candidates of joining the ring current.

We also analyze the kinematics of H+ ions using an ensemble of 1000 particles with varying r(t0) and Ek(t0) for the same input as above. In Fig. , we plot the final kinetic energy and pitch angle as a function of the initial values. The overall behavior resembles that of the (heavier) O+ ions; notice, for example, the regions of quasiperiodic and quasiballistic motion in Fig. c, similar to the ones in Fig. c. Nevertheless, the effect of acceleration, as imprinted in Fig. a, is found to be much weaker. This is connected to the difference in charge / mass ratio of the different ion species, and verifies the known storm-time composition for the energy density of the ring current, which is dominated by the O+ ions coming from the plasma sheet, in contrast to the situation in quiet time where H+ is the majority species see, for example,. In Fig. c one observes that the regions of periodic-like pitch angle behavior are now more narrow, which is in accordance with the fact that hydrogen ions launched from the plasma sheet are not effective in assimilating into the ring current region.

Going one step further, we estimate the statistical weight of each one of the populations formed by distributing the ions launched from the plasma sheet to the types of orbits described in the above (ring current, near-Earth tail, precipitating and escaping), and the outcome is shown in Table 2. The simulations involved two different ensembles of oxygen and hydrogen ions with Nens= 10 000 particles each, which were injected from the plasma sheet with random initial conditions for r(t0) within [18RE,22RE], for Ek(t0) within [2,6] keV and for θ(t0) within [20,30]∘, and all the other input being the same as above. One should notice that the conclusions drawn from Figs.  and , on the basis of single-particle dynamics, are verified. We highlight that, according to the computations, 46 % of the O+ ions of the ensemble are incorporated into the ring current, in contrast to nearly 4 % of the H+ ions, whereas a little more than 20 % in both species occupy the near-tail region; however, 25 % of the O+ particles and 39 % of the H+ ones escape the inner geospace.

The magnetic perturbation and the connection of Dst to the ring curren, as well as the contribution of each current source during the event phases, are under debate. In many cases, Dst is assumed to be correlated with the ring current energy from storm maximum well into recovery, on the basis that ring current ions provide the primary contribution to the storm-time Dst depression . However, it is suggested that Dst is also related to other sources, the effect of which may become important during disturbances. Based on ground measurements, indicate circumstances where the tail currents dominate the Dst development during storms. assess the effect of the tail currents on Dst by introducing a correction to the total current density, in terms of subtracting the magnetic curl in the tail regions as calculated by T89 and T96. The tail current was found to be most dominant in the end of the growth phase, and the accretion to Dst was estimated to scale up to 25 %.

A straightforward approach to calculate the Dst index from test particles involves the computation of the electric current densities from the particle velocities and the derivation of the generated magnetic fields (according to Ampere's law). However, the increased difficulty in the computation of surface current densities from particle orbits and the complexity of calculating the magnetic field from the electric currents, as well as the requirement of including the real positions of the ground-based sensors, imply a poor modeling performance. For studies related to the inner magnetosphere, the connection of Dst with the energy of the ring current has been described in terms of the Dessler–Parker–Sckopke (DPS) relation . The advantage here is that, at input, the kinetic energy of the local plasma is required, which is a scalar quantity and simple to deduct from the test-particle results.

The original DPS relation, which connects the energy Epp stored in a specific plasma population of the magnetosphere with the associated magnetic field perturbation, takes into account only those energetic particles that gyrate around the magnetic field lines and drift longitudinally due to the field gradient. In this context, the DPS formalism provides a sufficient estimation of the perturbations due to the ring current dynamics, as well as a well-balanced one of the near-Earth tail current contribution. With the introduction of a correction term for the magnetopause current in the original relation , one obtains a modified equation for the Dst index, Dst=-μ02πBERE3∑ppEpp+bdpsPdyn+cdps, where bdps is associated to the magnetopause correction and cdps to the quiet-time energy level. Equation () yields that, in order to estimate Dst, with the values of Pdyn, bdps and cdps for a specific event or scenario, one only requires the computation of Epp for the ring and tail plasma populations.

In the simulations, the ring current particles are assumed to be confined inside a torus with radii Rrc=6RE and rrc=3RE (i.e. extending from 3 RE to 9 RE), whereas the near-Earth tail region is defined as the remaining area in the simulation box ranging within [rrc,rrc+Rntl] along the Sun–Earth axis and [-Rntl,Rntl] in the other two directions, with Rntl=20RE. The energies Erc and Entl of the ring and near-tail current particles are described over the average energy of H+ and O+ test ions that belong to these currents. This is quantified by Epp=Vpp∑inpp,iEkpp,i, where npp,i is the plasma density of each ion species in each population and Vpp is the volumes of the regions occupied by the plasma populations, accordingly given by Vrc=2π2Rrcrrc2 and Vntl=Rntl3-Vrc. In the formula for Epp, the (ensemble) average value of the kinetic energy for the O+ and H+ ions in each current is computed, at each time step, over the particles contained inside the specific region at that time.

The results of the Dst computation using the scheme described above are presented in Fig. . The event scenario explored is again the one introduced in Sect. , i.e. a single disturbance that begins at t=0 from a quiet state with Kp =1, reaches its maximum level Kp=5 at t=30 min and returns to the quiet state until t=35 min. Two thousand test particles were used for the computation, divided in two different ensembles: one of 1000 oxygen ions launched from the plasma sheet, and one of 1000 hydrogen ions started in the ring current. The initial conditions for the MLT and the pitch angle were the same for both species, ϕ(t0)=24 h and α(t0)=90∘, whereas the initial radii, latitudes and kinetic energies were assigned randomly within different ranges for each species: for O+, r(t0) ranged in [18RE,22RE], θ(t0) in [20,30]∘ and Ek(t0) in [2,6] keV, while for H+ the corresponding intervals were [5RE,7RE], [0,5]∘ and [1,3] keV. The test ions were traced from 8 min before the beginning of the event until t=120 min, and at each time step the Dst index was computed from Eq. () and associated relations, where it was assumed that Pdyn=4 nPa, bdps=7.26 nT/nPa1/2, cdps=-11 nT, nrc,O=nrc,H=10 cm-3 and nntl,O=nntl,H=1 cm-3.

In Fig. a we plot the dynamic evolution of Dst, and the contributions from the ring current and the near-Earth tail plasma distributions to its value are given for comparisons. A qualitative agreement with the usual evolution of Dst during a substorm is seen: during the growth phase, Dst decreases continuously, with the most rapid variation being around the interchange from growth to relaxation phase, and values of Dst indicating magnetic perturbation persist for some time after the event termination. Overall, the contribution of the ring current to Dst is larger than the one coming from the energetic particles in the near-Earth tail region. The contribution of the tail current is measurable up to t=1.2 h, with a maximum near the end of the event growth (t=0.55 h), and from there on the Dst is essentially determined only by the ring current; this is in agreement with the behavior stated in and . The contribution to Dst by the tail current is found equal to 30 % on the average. This is a little larger than the reported figure of 25 % in the literature; however, such deviations are justified considering the adopted assumptions in these models.

A comparison of Kp and Dst during geomagnetic events is necessary for assessing their differences in response to different storm-time current systems. In cases where the dynamic pressure and the IMF both refer to the same category in storm magnitude, the minimum Dst is expected to decrease as a function of Kp. This has been verified in terms of an additional computation, where the maximum value of Kp during the event, attained right at the end of the growth phase, was modified from 1 to 7 (these are the minimum and maximum disturbance levels allowed by the T89 model) and, in each case, the minimum Dst value was recorded.

The correlation of the maximum values of |Dst| and Kp is shown in Fig. b and, as expected, has an increasing monotony. In the same figure, our result is compared to the linear regression curves derived from the statistical evaluation of data from substorm events during 1987–1996 and storms in the period 1996–1999 . The comparison with the results of shows an agreement only in the range of values 4< Kp <5, and with only for Kp >5+, which correspond to moderate and intense events. The main sources of disagreement in the other ranges are probably connected to the difference of the reference values of the geomagnetic disturbance (dynamic pressure, IMF, plasma density) in the analyzed data with respect to the input given to the code, as well as to the differences with the corresponding values in the data set employed by the T89 model.

In this paper, we employ a collection of models for the electric and magnetic field in the inner magnetosphere for the investigation of the dynamic evolution of the ring current and the near-Earth tail ion population during the occurrence of magnetospheric disturbances. Within this research framework, we have developed an orbit-solving code which computes the test-particle motion due to convection, corotation and Faraday induction in the dynamic magnetic and electric fields of the magnetosphere. We have used the code to study the ion dynamics, and in particular the dependence of ion acceleration on the initial conditions. Furthermore, we performed a numerical estimation of the Dst index based on the test-particle energies. The results of all computations have been found to be in qualitative agreement with previous studies on the ring current evolution during magnetospheric activity.

The ion motions have been traced by solving the nonrelativistic Lorentz equation, without adopting simplifications at any stage of the computation. In practice, one usually shifts to the GC equations when the particle reaches a distance smaller than a threshold radius, from where on the GC approximation is empirically assumed to be valid. In this respect, the choice of retaining the full-orbit description prevents inaccuracies from occurring in cases where some of the adiabatic invariants are broken. During intense disturbances, such cases have the potential to occur locally in space/time, and we have verified this situation by finding major deviations in the computation of a specific trajectory for several values of the threshold distance.

The analysis of test-particle orbits reveals fragments of the ion dynamics during the disturbance. We have identified three main types of ion orbits: orbits getting trapped around Earth, orbits precipitating in the Earth's atmosphere, and others escaping from the inner geospace. During the event, a percentage of oxygen ions launched from the plasma sheet are found to be accelerated and become trapped in the ring current. However, hydrogen ions (which are known to populate the ring current during quiet times), mainly escape from the inner geospace when launched from the plasma sheet. The largest part of the O+ acceleration occurs during the relaxation phase, where the magnetic field exhibits a steep decrease and, consequently, the induced electric field attains large values. The addition of this component to the convection field provides a mechanism for the observed energization levels of ions which drift towards the ring current region, contrary to an electric field purely due to plasma convection a similar result was found in.

Further analysis of the ion motions reveals a sensitive dependence of the particle dynamics on the initial conditions. We have found regions in geospace, including the plasma sheet, from where injected oxygen ions get preferentially accelerated, while ions starting from other regions may or may not appear a net energization depending on the initial energy. For O+ launched from the plasma sheet, the possibility for acceleration is found to be larger for ions having low energy at the beginning of the event. Consequently, the composition of the ring current may be modified by oxygen ions, the majority of which are initially in specific phase-space regions, which get accelerated and drift towards Earth. These findings are consistent with the results of previous studies on the role of substorms on the ring current dynamics, and have been verified here by an additional simulation.

For the effect of each current source to the Dst index during the event phases, we have concluded that one should, in principle, also account for the dependence of Dst on other effective sources apart from the ring current energy. Our computation of the Dst, in terms of the Dessler–Parker–Sckopke relation and test-particle results, indicates a measurable contribution from the near-Earth tail current of 30 % on the average, and yields a fair agreement with other estimations indicated in the literature (∼ 25 %). In the course of the event, the largest contribution of the tail current occurs during the growth phase, and persists for some time past its maximum. Thereafter, the effect of the tail currents gradually fades away, and the value of Dst is driven only by the ring current. Dst retains small values (related to meaningful disturbances) for long times after the event termination. A more accurate estimation of Dst may be achieved with the inclusion of the physics of loss mechanisms (collisions, cyclotron emission) and wave–particle interactions.

The present work may serve as the final link in a Sun-to-Earth modeling chain of major solar eruptions, providing an estimation of the inner geospace response once the solar burst reaches Earth. In this frame, a comparison of our results with other available models e.g.,, as well as with data from observations, may act as further validation. In a relevant work by , a benchmark of models was conducted, and the results showed that the computed ring current, for moderate and intense disturbances, depends on the field models. In our work, the benchmark of the T89 model has shown few differences within the simulated region in comparison to the later models, the facilitation of which may, however, increase the accuracy in the magnetic field. Also, the Kp  index is used as a parameter to describe the geomagnetic field during the disturbances, which is partially in contrast to the global character of the specific index see, for example,. In order to assess our results, the correlation of Kp and Dst was followed and a connection was found with previous studies.