Potential Field Source Surface PFSS Models

PFSS models are perhaps the most useful Sun-heliosphere connection models to date and have many applications (Arge et al., 2002; Luhmann et al., 2002; DeRosa and Schrijver, 2002). They use the potential field assumption mentioned above and hence decouple the coronal heating problem and the open-flux connection problem by making the inner corona exclusively magnetically dominated and time-independent. For a given choice of source surface radius, rs , the magnetic field potential then can be written as a sum of spherical harmonics.

The PFSS models reveal important characteristics of the Sun-heliosphere connection, which have been successfully confirmed. The first conclusion from this model is also demonstrated in Figure 2, namely that open flux organizes itself into volumes of open field with a given polarity. The interface between these regions of given polarity is a current layer which extends from the outer corona into the heliosphere to form a global heliospheric current sheet. Figure 2 shows the solar magnetic field and the heliospheric extension of this current sheet for two specific conditions. The top-left panel shows photospheric magnetic fields at solar minimum, for Carrington rotation 1913. The top-right panel shows the global structure of the heliospheric current sheet arising from this field configuration. Clearly, the dipole moment is the most dominant part of multipole expansion. The dipole is slightly tilted and also shows some additional warping, which can lead to multiple crossings of a spacecraft at a given heliospheric latitude. The bottom panels of Figure 2 show the equivalent figures near solar maximum conditions, for Carrington rotation 1969. Again, there is only one single-current sheet with interesting characteristics. First, the current sheet extends over all latitudes and shows a large

Figure 2. Photospheric distribution of magnetic field, from Wilcox solar observatory, and corresponding current-sheet orientation for Carrington rotations 1913 and 1969. Both solar minimum and solar maximum conditions lead to a single current sheet which extends to high latitudes near solar maximum. Computations from Riley et al. (2002).

Figure 2. Photospheric distribution of magnetic field, from Wilcox solar observatory, and corresponding current-sheet orientation for Carrington rotations 1913 and 1969. Both solar minimum and solar maximum conditions lead to a single current sheet which extends to high latitudes near solar maximum. Computations from Riley et al. (2002).

degree of warping induced from higher-order multipoles that become dominant during this period of the solar cycle.

Model predictions for the location of the current sheet have been successfully tested throughout the previous solar cycles, using in-ecliptic measurements near Earth, but also, for the first time, using high-latitude measurements from Ulysses, as indicated in Figure 1. These combined measurements were shown to be in remarkable agreement with predictions. The models have also been used successfully to describe the magnetic topology of the closed corona (DeRosa and Schrijver, 2002), even though deviations from potential fields are discernable from detailed observations (Schrijver and van Ballegooijen, 2005). PFSS models, without any additional correction terms, also predict a latitude-dependent magnitude of the total magnetic flux emerging into the heliosphere, which is inconsistent with observations (Smith et al., 2001). Such correction terms were proposed by several authors (Wang and Sheeley, 1995 and references therein), but are generally not included for practical use, such as the computation of magnetic field expansion factors.

As discussed above, PFSS models are generally unsuitable to describe the time evolution of the global magnetic field. They have nonetheless been used in the fashion of cartoons, "many images make a movie," far beyond the realm of physical applicability of PFSS models. This approach, of course, neglects crucial physics, but its physical meaning can be shown to be equivalent to a situation in which the timescale for magnetic dissipation, or reconnection effects, rRec, outpaces the typical timescale of the evolution of this structure, rEvo, or, rRec ^ rEvo-

Under these assumptions, the evolution of the heliospheric current sheet from one solar minimum to the next is well approximated by a slow, but highly irregular rotation of this current sheet. During its 11 years (on average), the current sheet simply rotates through 180° to complete its field reversal.

It should be pointed out that there may still be benefits to the application of PFSS models in such a time-dependent mode, as shown by Arge and Pizzo (2000), with respect to the predictions of the polarity of the heliospheric magnetic field and the solar wind speed. These physical quantities appear to be determined by large-scale topology successfully modeled by PFSS models. But, this approach is limited: Large-scale field-deviations from the average Parker model, for example, result from evolutionary aspects of the field, and cannot be derived from a PFSS approach. The knowledge of this large-scale field is required for the prediction of energetic particles, for example.

Some MHD models, such as Riley et al. (2002), predict multiple disconnected current sheets during sole Carrington rotations. Thus far this has not been directly observed (Smith et al., 2001), but we have limited data. We should also remember the important model assumptions, and the limitations, of observing magnetic fields only from near the Earth, which clearly limits the knowledge of the boundary conditions for any heliospheric model calculations. However, their fundamental shortcomings are their neglect of plasma time dependence and plasma interactions, which have to be addressed in a more rigorous way.

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