Our present day understanding of the Sun’s open flux comes mainly from an important result from the Ulysses mission: the magnitude of the radial IMF component in the heliosphere is independent of latitude (Balogh et al., 1995). Thus, magnetometer measurements at a single location at 1 AU may be used to deduce the total open flux simply by multiplying by the surface area of a sphere. In Figure 13, the variation of open flux relative to sunspot number can be seen for the last 3.5 solar cycles. Over a solar cycle the open flux varies at most by a factor of two, even though the surface flux shows a much larger variation. However, this variation is not regular from one cycle to the next. It is clear from the graph that Cycles 21 and 22 have a much larger variation in open flux than Cycles 20 and 23, indicating a complex relationship between the surface and open flux. A key property of the open flux is that it slightly lags behind the variation in sunspot number and peaks 1 – 2 years after cycle maximum. The peak time of open flux therefore occurs around the same time as polar field reversal, indicating that contributions from both polar coronal holes and low-latitude coronal holes must be important.
While both direct and indirect observations exist for the Sun’s open flux, such measurements cannot be made for other stars. Due to this, theoretical models which predict the magnitude and spatial distribution of open flux are used. Within the stellar context, the distribution of open flux with latitude (McIvor et al., 2006) plays a key role in determining the mass and angular momentum loss (Cohen et al., 2007) and subsequently the spin down of stars (Weber and Davis Jr, 1967; Schrijver et al., 2003; Holzwarth and Jardine, 2007).
Over the last 20 years a variety of techniques has been developed to model the origin and variation of the Sun’s open flux. One category are the semi-empirical magnitude variation models. These follow only the total open flux, and are driven by observational data. Different models use either geomagnetic data from Earth (Lockwood et al., 1999, 2009), sunspot numbers (Solanki et al., 2000, 2002), or coronal mass ejection rates (Owens et al., 2011). Such models have been used to extrapolate the open flux back as far as 1610 (in the case of sunspot numbers), although the results remain uncertain. However, these models consider only the total integrated open flux, not its spatial distribution and origin on the Sun. To describe the latter, coupled photospheric and coronal magnetic field models have been applied. The photospheric field is specified either from synoptic magnetic field observations or from magnetic flux transport simulations (Section 2.2), and a wide variety of coronal models have been used. These range from potential field source surface (PFSS, Section 3.1) models to current sheet source surface (CSSS, Section 3.1) models and more recently to nonlinear force-free (NLFF, Section 3.2) models. In all of these coronal models, field lines reaching the upper boundary are deemed to be open, and the total (unsigned) magnetic flux through this boundary represents the open flux.
Wang and Sheeley Jr (2002) combined synoptic magnetograms and PFSS models to compute the open flux from 1971 to 1998. The synoptic magnetograms originated from either Wilcox Solar Observatory (WSO, 1976 – 1995) or Mount Wilson Observatory (MWO, 1971 – 1976, 1995 – 1998). The authors found that to reproduce a good agreement to IMF field measurements at 1 AU, they had to multiply the magnetograms by a strong latitude-dependent correction factor (). This factor, used to correct for saturation effects in the magnetograph observations, significantly enhanced the low latitude fields compared to the high latitude fields but gave the desired result.
The use of such a latitude dependent correction factor was recently questioned by Riley (2007), who pointed out that the correction factor used by Wang and Sheeley Jr (2002) was only correct for use on MWO data and did not apply to WSO data. Instead WSO data, which made up the majority of the magnetogram time series used, should have a constant correction factor of 1.85 (Svalgaard, 2006). On applying this appropriate correction and repeating the calculation, Riley (2007) showed that PFSS models gave a poor fit to observed IMF data (see Figure 4 of Riley, 2007). To resolve the difference, Riley (2007) put forward an alternative explanation. He assumed that the open flux has two contributions: (i) a variable background contribution based on the photospheric field distribution on the Sun at any time, such as that obtained from the PFSS model, and (ii) a short term enhancement due to interplanetary coronal mass ejections (ICMEs), for which he gave an order-of-magnitude estimate. Through combining the two a good agreement to IMF field observations was found.
As an alternative to synoptic magnetic field observations, magnetic flux transport models have been widely used to provide the lower boundary condition in studies of the origin and variation of the Sun’s open flux. A key element in using magnetic flux transport simulations is that the distribution and strength of the high latitude field are determined by the properties and subsequent evolution of the bipoles which emerge at lower latitudes. Hence, assuming the correct input and advection parameters, the model will produce a better estimate of the polar field strength compared to that found in magnetogram observations which suffer from severe line-of-sight effects above 60° latitude. Initial studies (using a PFSS coronal model) considered the variation in open flux from a single bipole (Wang et al., 2000; Mackay et al., 2002a) as it was advected across the solar surface. Through doing so, the combined effects of latitude of emergence, Joy’s Law and differential rotation on the open flux was quantified. Later studies extended these simulations to include the full solar cycle, but obtained conflicting results.
To start with, Mackay et al. (2002b), using idealised simulations and commonly used transport parameters, found that such parameters when combined with PFSS models produced an incorrect time-variation of open flux. The open flux peaked at cycle minimum, completely out of phase with the solar cycle and inconsistent with the observed 1 – 2 year lag behind solar maximum. The authors attributed this to the fact that in PFSS models only the lowest order harmonics (see Section 3.1), which are strongest at cycle minimum and weakest at maximum, significantly contribute to the open flux. They concluded that the only way to change this was to include non-potential effects (see below) which would increase the open flux contribution from higher order harmonics during cycle maximum. In response, Wang et al. (2002b) repeated the simulation and found similar results when using bipole fluxes and parameters deduced from observations. However, they were able to obtain the correct variation in open flux by both trebling all observed bipole fluxes and increasing the rate of meridional flow to 25 m s–1 (dash-dot-dot-dot line in Figure 3b). These changes had the effect of significantly enhancing the low latitude field at cycle maximum but weakening the polar fields at cycle minimum. As a result the correct open flux variation was found. Opposing such a strong variation in the input parameters, Schüssler and Baumann (2006) put forward another possibility. Through modifying the magnetic flux transport model to include a radial decay term for (Section 2.2), and changing the coronal model to a CSSS model with and (Section 3.3), they found that the correct open flux variation could be obtained, but only if new bipole tilt angles were decreased from to . Later studies by Cameron et al. (2010) using the same technique showed that the radial decay term was not required if the tilt angles of the bipoles varied from one cycle to the next, with stronger cycles having weaker tilt angles.
The discussion above shows that, similar to the polar field strength (Section 2.2), the correct variation of the Sun’s open flux may be obtained through a variety of methods. These include variations in the bipole tilt angles, rates of meridional flow, and different coronal models. More recently, Yeates et al. (2010b) showed that standard values for bipole tilt angles and meridional flow may produce the correct variation of open flux if a more realistic physics-based coronal model is applied (see Section 3.2.3). By allowing for electric currents in the corona, a better agreement to the IMF measurements can be found compared to those from PFSS models. Figure 14a compares various PFSS extrapolations using different magnetogram data (coloured lines) to the measured IMF field (grey line). In this graph the discrepancy between all potential field models and the IMF is particularly apparent around cycle maximum. To resolve this, Yeates et al. (2010b) ran the global non-potential model described in Section 3.2.3 over four distinct 6-month periods (labeled A – D in Figure 14a). Two periods were during low solar activity (A and D) and two during high solar activity (B and C). The results of the simulation for period B can be seen in Figure 14b. In the plot, the dashed lines denote open flux from various PFSS extrapolations, the grey line the observed IMF field strength and the black solid line the open flux from the non-potential global simulation. This clearly gives a much better agreement in absolute magnitude terms compared to the PFSS models. Yeates et al. (2010b) deduced that the open flux in their model has three main sources. The first is a background level due to the location of the flux sources. This is the component that may be captured by PFSS models, or by steady-state MHD models initialised with PFSS extrapolations (Feng et al., 2012). The second is an enhancement due to electric currents, which results in an inflation of the magnetic field, visible in Figure 14b as an initial steady increase of the open flux curve during the first month to a higher base level. This inflation is the result of large scale flows such as differential rotation and meridional flows along with flux emergence, reconfiguring the coronal field. A similar inflation due to electric currents enhances the open flux in the CSSS model, although there the pattern of currents is arbitrarily imposed, rather than arising physically. Finally, there is a sporadic component to the open flux as a result of flux rope ejections representing CMEs similar to that proposed by Riley (2007). While this model gives one explanation for the open flux shortfall in PFSS models, we note that alternative explanations have been put forward by a variety of authors (Lockwood et al., 2009; Fisk and Zurbuchen, 2006). Again, although models have been successful in explaining the variation of the open flux, there is still an uncertainty on the true physics required. Free parameters that presently allow for multiple models must be constrained by future observations. At the same time, it is evident from using existing observations that care must be taken in their use. In particular, the effect on the modeling results of practical limitations such as calibration, resolution, instrumental effects, and data reduction effects must be quantified.
Although they include the spatial distribution of open flux, the models described above have mainly focused on the total amount of open flux, because this may be compared with in situ measurements. However, these models can also be used to consider how open flux evolves across the solar surface, and its relationship to coronal holes. Presently, there are two opposing views on the nature of the evolution of open flux. The first represents that of Fisk and Schwadron (2001) who suggest that open flux may diffuse across the solar surface through interchange reconnection. In this model, the open flux may propagate into and through closed field regions. The second view represents that of Antiochos et al. (2007) who postulate that open flux can never be isolated and thus cannot propagate through closed field regions. Results of global MHD simulations testing these ideas are discussed in the next section.
Living Rev. Solar Phys. 9, (2012), 6
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