Early Development of the Ring Current Model

The ring current model hosted in MSFC is a kinetic model that solves a Boltzmann initial/boundary value problem with specified electric and magnetic fields. Rather than tracking particles trajectories in detail, a bounce-averaged approach is taken. The bounce-averaged kinetic equation of ring current species a, including the effects of charge exchange losses and Coulomb drag on energetic particles interacting with hydrogen geocorona and the plasmasphere, respectively, can be written as,

(1)

where is the average phase space distribution function along the field line between mirror points, Ro is the radial distance at the equator, f is the magnetic local time, v is velocity, sa is the cross section for charge exchange of species a with the neutral hydrogen, nH is the hydrogen density, M is the magnetic moment, is the rate of change of M due to Coulomb interactions with the thermal plasmas, and is the bounce-averaged value of quantity x. An additional loss term (last term on the right hand side of (1)), with a lifetime of half of the bounce period, is applied to particles in the loss cone, which is defined at 800 km. Since we are considering particles (1 E 300 keV) with bounce periods much shorter than the decay lifetimes, fa is assumed to be constant along field line and thus can be replaced by the distribution function at the equator. A full description of this ring current model is given in Fok et al. [1993, 1995a].

In the early version of the model, a dipole magnetic field has been used for the region between 2.0 and 6.5 RE. The circulation electric field is prescribed by the Volland-Stern model, and the plasmasphere is from the model of Rasmussen et al. [1993], with variations prescribed by the time series of Kp values for any specific storm interval. The initial and outer boundary conditions are derived from observations of hot plasmas, e.g., those of AMPTE/CCE. The main contribution of this ring current model lies in its attention to the pitch angle-dependent effects of charge exchange and Coulomb collisions. It computes the equatorial pitch angle distributions as a function of ion species and location in the equatorial plane, leading to an effective 3D description of the hot plasma distribution in space.

This model has been extensively compared with observations, validating the essential features of the model, as illustrated by example in Figures 1 and 2.

Figure 1. The comparison of calculated H+ pitch angle-averaged differential flux (solid lines) with AMPTE/CCE observations (data points) of May 2, 1986 storm.

Figure 2. The comparison of calculated pitch angle distribution (solid lines) with AMPTE/CCE measurements (data points). Dashed lines in the bottom panels result from inclusion of pitch angle diffusion with Daa = 5¥10-6 s-1.

As Figures 1 and 2 amply illustrate, the model exhibits good general agreement in both energy distributions and pitch angle distributions with direct observations of ring current plasmas. Where discrepancies exist, they often result from the need for a treatment of wave-driven pitch angle diffusion and/or treatment of ions outflow from the ionosphere into the inner magnetosphere.

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