We investigate the effect of magnetic disturbances on the ring current buildup and the dynamics of the current systems in the inner geospace by means of numerical simulations of ion orbits during enhanced magnetospheric activity. For this purpose, we developed a particle-tracing model that solves for the ion motion in a dynamic geomagnetic field and an electric field due to convection, corotation and Faraday induction and which mimics reconfigurations typical to such events. The kinematic data of the test particles is used for analyzing the dependence of the system on the initial conditions, as well as for mapping the different ion species to the magnetospheric currents. Furthermore, an estimation of Dst is given in terms of the ensemble-averaged ring and tail currents. The presented model may serve as a tool in a Sun-to-Earth modeling chain of major solar eruptions, providing an estimation of the inner geospace response.

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

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

There are cases where global fluid and magnetohydrodynamic (MHD) simulations
reproduce the observed changes in the magnetic topology to quite good
accuracy

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.

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

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.

The magnetic field in geospace is expressed as the sum of two
contributions: the first one is from the Earth's terrestrial field, whereas
the second comes from the external field generated by the electric currents
flowing inside the magnetosphere (including the magnetopause). The Earth's
magnetic field is well approximated as the one of a tilted dipole magnet
with inverse polarity

The second component of the geomagnetic field, denoted by

The mainstream of models for the static part of

For the visualization of the magnetic field, one employs the standard
field-line tracing technique

Geomagnetic field map on the GSM

For proper application of the Tsyganenko models, the role of the
differences between the models and the properties of the computed physics,
especially in strongly disturbed cases, has to be investigated. The benchmark
of these models against observations is an issue that has been addressed by a
number of authors.

A comparison of T89, T96 and TS05 is presented in Fig.

Input to the Tsyganenko models T89, T96 and TS05 for the benchmark case
computations presented in Fig.

Computation of the relative deviation between the computations of
the geomagnetic field, for zero tilt angle in a disturbed magnetosphere
(Kp

The dynamic part of the magnetic field is determined by the modification of
the geomagnetic parameters in time. With introduction of the vector

In our model, an event starts at time

The electric field is divided into three components

Function

The effect of the model differences to the computed dynamics in the
transition to stormy conditions is again put under question. In

From the aforementioned tools we choose to employ the VSMC model, which
combines sufficient accuracy in the physics description with simplicity in
the computer implementation

Vector fields coming from a scalar potential are represented in terms of their
equipotential (contour) surfaces. The contour surfaces of

Contours of the convection and corotation electric potential over
the GSM

The role of the electric field component induced by the dynamic variation in
the magnetic field in properly modeling the solar-driven perturbations is
very important. This is due to the fact that it has a short space/timescale,
which is effective in accelerating ions to very high energies (as observed
during storms and substorms), whereas the convection process forms a
distribution of plasma currents of comparatively low energy. In this context,
the total electric field is expressed in terms of the potentials

The test-particle model computes the near-Earth ion dynamics
during the geomagnetic disturbance by following the 3-D trajectories under
the effect of the associated electric and magnetic fields. The particle
trajectory is traced by solving numerically the Lorentz equation including
the gravitational force

The particle motions may also be evaluated in terms of a reduction to the
full model, depending on the validity of the guiding-center (GC)
approximation over the simulated region. The GC trajectory describes the
overall motion well in cases where the electric/magnetic field variations remain
sufficiently small over each revolution

The particle-tracing scheme combines the models presented so far: T89 is
employed for the static part of

The orbits may be traced with the Lorentz equation, with no simplification
adopted at any stage of the computation. The GC model, in the regions where
it is valid according to the conditions (Eq.

In the literature, the benchmarking between the different magnetospheric
particle solvers in the presence of intense electric and magnetic fields is
not sufficiently extensive and the results are contradictory.

Application of the model interchanging technique in test-particle
computations for different values of the threshold radius:

In order to clarify this issue, we compute a specific trajectory for several
values of the threshold distance

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

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.

Projections of a trapped O

In Fig.

Three-dimensional plot of ion orbits which conclude outside the inner
magnetosphere, with initial conditions same as in Fig.

Regarding the particle acceleration, in Fig.

Dynamic evolution of the

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

In Fig.

We also analyze the kinematics of H

Final kinetic energy of

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

Distribution to the ring current (

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

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 original DPS relation, which connects the energy

In the simulations, the ring current particles are assumed to be confined
inside a torus with radii

The results of the Dst computation using the scheme described above are
presented in Fig.

In Fig.

Analysis of Dst based on an ensemble of 2000 test ions, 1000 O

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

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

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

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 (

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

The authors would like to thank L. Vlahos and A. Metallinou for the useful discussions. This research was co-financed by the European Union (European Social Fund) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework Research Funding Program: Thales. Investing in knowledge society through the European Social Fund. The topical editor, C.-P. Escoubet, thanks the two anonymous referees for help in evaluating this paper.