#### 4.1. Model Description

[16] The Rice Convection Model (RCM) self-consistently calculates the inner magnetospheric particle distribution, the Region 2 Birkeland currents, and resulting electric field patterns. Magnetic field variations caused by changes in the Region 2 currents and the particle distribution are not presently calculated, but this treatment is presently being developed. The RCM solves the fundamental equation of magnetospheric-ionosphere coupling [*Vasyliunas*, 1970; *Wolf*, 1983]

where ∇_{H} is the horizontal gradient in the ionosphere, is a tensor representing the field-line-integrated ionospheric conductance (one hemisphere), Φ is the electric potential in the solar frame with the corotation field removed, *I* is the magnetic dip angle, *B*_{i} and *B*_{eq} represent the magnetic field strengths in the ionosphere and equatorial plane, _{eq} is the unit vector of the magnetic field direction in the equatorial plane, *p* is the pressure, and *V* = ∫ *ds*/*B* is the volume of a magnetic flux tube with one unit of magnetic flux.

[17] Assuming strong, elastic pitch-angle scattering, the energy invariant

is conserved, where *E* is particle energy. The energy spectrum is divided into a discrete set of energy invariants λ_{s}. The number of particles per unit magnetic flux of a given energy invariant λ_{s} is denoted η_{s}. Here η_{s} is called the density invariant and is conserved along a drift path, except for the effects of loss:

where _{drift,s} is bounce-averaged drift velocity and τ_{s} is the loss lifetime. Particle sources are neglected in the runs described in this paper. Ion losses are neglected but effects of electron precipitation are included. The ideal monatomic-gas adiabatic invariant *pV*^{5/3} is related to the drift invariants by

The drift velocity is given by

[18] This RCM run includes two chemical species, electrons and singly charged ions. Since *H*^{+} and *O*^{+} ions of given energy drift the same and we are neglecting ion loss, there is no need to keep separate track of *H*^{+} and *O*^{+}. The distributions of electrons and ions are each divided into 20 separate energy invariant species.

[19] The code represents the ion population in terms of contours of constant η_{s}. Since η_{s} is constant along an ion drift path, the contours move at the drift velocity given by equation (5). Each contour is therefore defined by a set of drifting test particles. For each chemical species and each invariant energy level, ten contours levels are used. Nine levels define the initial trapped plasma distribution. The tenth level, representative of the central plasma sheet, is set to correspond with the plasma outer-boundary condition, which is uniform on the boundary and constant in time. This contour-based representation is used because it produces no numerical diffusion and allows a clean separation between particles that were in the original trapped distribution and those that came from the plasma sheet during the storm.

[20] Plasma sheet electrons are considered differently than ions. The electron population is specified in terms of η_{s} values at grid points rather than a series of test particles. The use of this grid-based representation allows the RCM to incorporate electron loss. Kilovolt electrons have lifetimes of only a few hours, so loss cannot be neglected [*Axford*, 1969]. As the electron plasma sheet convects Earthward, electron density decreases due to loss. In this RCM run, the electrons are lost by strong pitch-angle scattering, reduced by a factor of 2/3 for high Kp (greater than 4+) or a factor of 1/3 for lower Kp [*Schumaker et al.*, 1989].

#### 4.2. Model Inputs

[23] The basic inputs to the RCM are the initial and boundary plasma distributions, the magnetic field, the background and auroral zone ionospheric conductances, and the potential on the poleward boundary. The total strength of the imposed large-scale convection is given by the polar cap potential drop, and is distributed along the ionospheric poleward boundary as sin(πMLT/12). For this study, we used the maximum DMSP-calculated potential drop in a given hour (shown in Figure 4). The magnetic field at a given time step is interpolated between Hilmer-Voigt magnetic field models [*Hilmer and Voigt*, 1995] specified every 15 min according to the calculated standoff distances and the Dst and ABI indices (shown in Figure 3). The dipole tilt is assumed to be zero.

[24] The initial ion distribution is perhaps the most complex input for this run. The preexisting ion population is inferred from MICS observations at different L-shells prior to the storm. The measured differential particle flux *j* is converted into the density invariant η by

where Δλ is the separation between energy invariant channels. A piecewise logarithmic interpolation between the MICS energies is used to specify the fluxes at the RCM energy invariants. The upper bound of the RCM energy invariant range is approximately 100 keV at geosynchronous orbit.

[25] In treating the initial trapped magnetospheric particle distribution, we consider only ions. Of course, electrons must be present in the initial distribution to balance the ion charge, however these electrons are assumed to have sufficiently low energies that they do not contribute significantly to the pressure gradients that drive the Birkeland currents; thus they need not be included explicitly in the RCM.

[26] The plasma sheet population, which enters the model as a boundary condition, is based upon published plasma sheet statistics. *Borovsky et al.* [1998a] found that plasma sheet conditions (denoted *ps*) are related to solar wind (denoted *sw*) conditions at the magnetopause. In particular, the number density of ions in the plasma sheet between 17.5 and 22.5 *R*_{e} downtail is given by

where the number densities are in cm^{−3}. The ion temperature is given by

where the plasma sheet temperature is in keV and the solar wind velocity is in kilometers per second. In this run, the plasma distribution η(λ) is assumed to be independent of time and position on the tailward boundary of the RCM. To set this distribution, the boundary plasma moments are set to *n*_{e} = *n*_{i} = 0.148 cm^{−3}, *T*_{i} = 8.32 keV, and *p*_{i} = 0.2 nPa at about 20 *R*_{E}. The 8.32 keV value was estimated from equation (7b) and the solar wind velocity measured at 0600 UT on 5 June (630 km/s), during the largest ring current injection. Use of the measured solar wind density (37.6 cm^{−3}) in equation (7a) leads to a plasma sheet density of 0.74 cm^{−3}, but that leads to unrealistically high flux levels at geosynchronous orbit. For that reason, we divide the plasma sheet density derived from equation (7a) by five (a rough factor determined from a series of model runs) in setting our outer boundary condition. This is a symptom of the pressure balance inconsistency [e.g., *Erickson and Wolf*, 1980; *Spence et al.*, 1989; *Borovsky et al.*, 1998b]. Standard magnetic field models imply that the adiabatic quantity *pV*^{5/3} decreases Earthward between about 20 *R*_{e} and 6.6 *R*_{e}, by processes that are not yet known and are not included in the RCM.

[27] The total electron density in the plasma sheet is set equal to the total ion density. The electron temperature is by *T*_{i}/*T*_{e} = 7.2 to agree with *Baumjohann et al.* [1989]. The density and temperature are converted into an density invariant-energy invariant spectrum using a Kappa function of the order of 6 [*Vasyliunas*, 1968; *Christon et al.*, 1989, 1991] where the density invariant is given by equation7

Here, *m* is the particle mass, *k* is the Boltzmann's constant, κ = 6 is the exponent of the high energy differential flux, and *a* = κ − 1.5. The inner edge of the plasma sheet is initially placed so that it just touches the modeling boundary at 1200 MLT and follows a contour of constant flux tube volume; thus the initial inner edge produces no Birkeland current.

[28] The final RCM inputs are the background conductance (produced by solar EUV ionization) and auroral enhancement (produced by particle precipitation). To compute the field-line-integrated background conductance, electron and ion densities and temperatures are taken from the International Reference Ionosphere (IRI) [*Bilitza*, 1997] and neutral densities from the Mass-Spectrometer Incoherent Scatter (MSIS) [*Hedin*, 1991] models. The background conductance is assumed not to change during the RCM simulation. In contrast, the auroral conductance changes during the RCM run, reflecting the RCM-computed electron distribution. The auroral conductances are adjusted at each time step to agree with the location and concentration of the electron plasma sheet. The precipitating electron flux is adjusted so that the integral of the electron precipitation flux over the RCM's magnetic latitude range agrees with the corresponding integral from the *Hardy et al.* [1985] model at each magnetic local time. Thus the latitudinal distribution of electron precipitation is forced to line up with the computed electron plasma sheet, and the total precipitation rate at each local time is realistic. The conductance algorithms have been discussed in more detail by *Sazykin* [2000].