A major issue with time-dependent modeling is what to assume for the time dependence of the diffusion coefficients, which is significantly more difficult to do from first principles than their energy or spatial dependence. A basic departure point (required to make progress) for the time dependence of the transport parameters to describe global long-term modulation is that propagating barriers (solar wind and magnetic field structures inhibiting the easy access of cosmic rays) are formed (and later dissipate) in the heliosphere during the 11-year activity cycle. The concept was first implemented in a model by Perko and Fisk (1983) and later extended by Potgieter and le Roux (1989). This is especially applicable to the phase of the solar activity cycle before and after solar-maximum conditions when large steps in the particle intensities have been observed. In fact, a wide range of interaction regions occur in the heliosphere, the largest being called global merged interaction regions (GMIRs) introduced by Burlaga and Ness (1993) and Burlaga, Perko, and Pirraglia (1993). They observed that a clear relation exists between cosmic ray decreases (recoveries) and the time-dependent decrease (recovery) of the magnetic field magnitude and extent local to the observation point. The paradigm on which these modulation barriers is based is that interaction (and rarefaction) regions form with increasing radial distance from the Sun. This happens when two different solar wind speed regions become radially aligned to form an interaction region when the fast one runs into the slower one, resulting in compression fronts with forward and backward shocks. When these relatively narrow interaction regions are extended and wrap almost around the Sun they are called corotating interaction regions (CIRs). Between 8 AU and 10 AU these CIRs begin to spread, merge, and interact to form merged interaction regions (MIRs).
Perko and Burlaga (1990) introduced MIRs in modeling as outward-propagating regions of enhanced magnetic field magnitude relative to the background field which then cause a localized region of decreased diffusion coefficients, acting in the process as diffusion barriers but also as drift barriers to the incoming cosmic rays. The latter was extensively modeled by Potgieter and le Roux (1994). Beyond 20 AU the MIRs merge to form GMIRs, which can become large in extent and capable of causing the large step-like changes in cosmic rays. Potgieter et al. (1993) found that the periods during which GMIRs affect long-term modulation depend on their rate of occurrence, the radius of the heliosphere, the speed with which they propagate, their spatial extent (and amplitude), especially their latitudinal extent (to disturb drifts), and the background modulation conditions (diffusion coefficients) they encounter. Drifts, on the other hand, dominate the solar-minimum modulation periods up to 4 years so that during an 11-year cycle a transition must occur (depending on how solar activity develops) from a period dominated by drifts to a period dominated by these propagating structures.
Equally important to long-term cosmic ray modulation are gradient, curvature, and current sheet drifts as confirmed by comprehensive modeling done by Potgieter et al. (1993) and le Roux and Potgieter (1995). They showed that it was possible to simulate, to the first order, a complete 22-year modulation cycle by including a combination of drifts, with time-dependent tilt angles, and GMIRs in a time-dependent modulation model. For reviews of their work, see Potgieter (1997) and references therein.
For recent contributions and appreciation of this process, see Zank and Muller (2003), Ferreira and Scherer (2006), and Florinski and Zank (2006).
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