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11 Models of Open Solar Flux Variation

The extension of the coronal field into the heliosphere and, hence, the modelling of the open solar flux variation, has been recently covered in another Living Review by Owens and Forsyth (2013). Therefore, as in the last section, only a brief review will be given here, to stress the extent to which the reconstructions are both feeding into the models and providing tests of them.

A number of theoretical concepts for the evolution of the heliospheric magnetic field have been proposed. Fisk (1999) argue that the Sun’s open flux tends to be conserved, with “interchange reconnection” (see Crooker et al., 2002) between open and closed solar fields resulting in an effective diffusion of open flux across the solar surface without, necessarily, any net change in the total open flux. In this case, the heliospheric field evolves with simple rotation of regions of positive and negative polarity separated by a single, large-scale heliospheric current sheet (Fisk and Schwadron, 2001; Jones et al., 2003). That this “Fisk circulation” can conserve the open flux does not mean there are not other processes that act simultaneously to cause it to grow and decay. It has been argued that emerging midlatitude bipoles cause closed coronal loops to rise and first destroy pre-existing open flux in the polar coronal hole (remnant from the previous solar cycle) and then build up a new polar coronal hole (of the opposite polarity) and so reverse the polar field of the Sun (Babcock, 1961; Wang and Sheeley Jr, 2003bJump To The Next Citation Point), which fits well with the migration of photospheric fields seen in magnetograph data (see the Living Review by Sheeley Jr, 2005). The evolution of the heliospheric magnetic field could also be facilitated by transient events (Low, 2001): specifically, Owens and Crooker (2006, 2007) and Owens et al. (2007) investigated the role of the magnetic flux contained in coronal mass ejections (CMEs) in the observed variation in flux seen by craft in the heliosphere. These different concepts are not mutually exclusive in many respects (see review by Lockwood, 2004). One complicating factor in this debate has been semantics: “open flux” in this review, and in many previous papers, is taken to be the same as coronal source flux; that is, the magnetic flux that leaves the solar atmosphere and enters the heliosphere by threading the coronal source surface at r = 2.5 R ⊙. As discussed in this review, it is a measurable quantity because of PFSS modeling (within the assumptions of that technique) and because of the Ulysses result allows the use of in situ magnetic field data. This is quite different from another definition of open flux which requires that it has only one footpoint still (effectively) attached to the Sun (e.g., Schwadron et al., 2008). Flux which appears to be in this category can sometimes be inferred for in-situ point measurements, for example, from heat flux or unidirectional strahl electron distribution functions (although scattering by heliospheric structure into other populations such as halo often makes this far from unambiguous) (Larson et al., 1997; Fitzenreiter et al., 1998; Owens et al., 2008b). Even if this could be done reliably, there is no way to quantify the total of such flux at any one time from such in situ point measurements. This is because there is no equivalent of the Ulysses result to generalize in situ point measurements into a global quantity. Lockwood et al. (2009c) have reviewed how various phenomena (coronal mass ejections, interchange reconnection, disconnection reconnection) influence both these two definitions of open flux.

The long-term change in the open flux (meaning coronal source flux, FS) deduced from geomagnetic activity has been reproduced by a number of numerical models of flux continuity and transport during the solar magnetic cycle, given the variation in photospheric emergence rate indicated by sunspot numbers (Solanki et al., 2000Jump To The Next Citation Point, 2002Jump To The Next Citation Point; Schrijver et al., 2002; Lean et al., 2002; Mackay and Lockwood, 2002; Wang and Sheeley Jr, 2002; Wang et al., 2002; Wang and Sheeley Jr, 2003b; Wang et al., 2005). The key fundamental principle was established by Solanki et al. (2000Jump To The Next Citation Point), namely the continuity of total open solar flux:

dFS ∕dt = S − L , (10 )
where t is time, S is the open flux emergence rate (the source term), and L is the total open flux loss term. This simple equation works without considering the precise distribution of open flux over the Sun and how it changes, although that will undoubtedly influence the loss rate greatly and could influence the source term as well. In order to extend the modelling back to the Maunder minimum, the group sunspot number, Rg has generally been used in some form to quantify S. An initial concern was that even a model as simple as Equation (10View Equation) may have too many free variables to be meaningful. However, it should be noted that the model was “trained”, and the coefficients defined, using the LEA99 open solar flux reconstruction that extended up to 1995. Therefore, although the first perihelion pass of Ulysses (between day 280 of 1994 and day 235 of 1995) was not independent data, it is significant that the model reproduced the open solar flux detected during the second perihelion pass (between day 353 of 2000 and day 301 of 2001, i.e., roughly half a solar cycle later) (Lockwood, 2003Jump To The Next Citation Point) and the third perihelion pass (almost a full solar cycle later). The model, therefore, has real predictive capability. Solanki et al. (2002Jump To The Next Citation Point) extended the modelling by adding more classifications of closed solar flux, each governed by its own continuity equations, to define the emergence rate S. This has been very useful in allowing centennial reconstructions of total and spectral solar irradiance which are, ultimately, constrained by the open solar flux reconstructions from geomagnetic activity.

Various forms have been used for the loss rate L. Solanki et al. (2000, 2002) and Vieira and Solanki (2010) used a constant fractional loss, i.e., L = FS∕τ where τ is the loss time constant. In addition, a constant absolute loss rate has been used by Connick et al. (2011) and a fractional loss rate that varies over the solar cycle by Owens et al. (2011aJump To The Next Citation Point). In particular, working from the assumption that the source term S was set by the CME emergence rate, Owens and Lockwood (2012Jump To The Next Citation Point) showed that the loss rate needed was cyclic over the solar cycle and was very well correlated with the tilt of the heliospheric current sheet, as predicted theoretically by Sheeley Jr and Wang (2001Jump To The Next Citation Point) and Owens et al. (2011a), and consistent with the observations that streamer belt disconnection events, as seen in coronograph images, tend to occur where the current sheet is tilted (Wang et al., 1999b,a; Sheeley Jr and Wang, 2001).

View Image

Figure 35: Long-term variation of unsigned open solar flux, 2F S. The grey area bounded by a black line is a model fit to the green line which is the LEA09 reconstruction. Ten year averages of cosmogenic isotope estimates of open solar flux are shown in blue (from 14C) and red (from 10Be), with solid and dashed lines showing linear and third-order fits of the heliospheric modulation potential variation to the open solar flux reconstruction from geomagnetic activity. Image reproduced by permission from Owens and Lockwood (2012Jump To The Next Citation Point), copyright by AGU.

Owens and Lockwood (2012Jump To The Next Citation Point) noted that during the long low minimum between cycles 23 and 24, CME flux emergence continued, and they postulated that this was a base-level emergence rate that would have continued at all times, including during the Maunder minimum. Using this they derived the modelled variation of open solar flux shown by the grey area Figure 35View Image. Note that the plot shows the unsigned open flux and so is 2FS. The modelled signed open solar flux at the end of Maunder minimum oscillated around a mean of about FS = 0.7 × 1014Wb. This is a bit larger than, but still consistent within uncertainties, with the estimate from the SEA10 reconstruction and Figure 29View Image of 14 (0.48 ± 0.29) × 10 Wb. The geomagnetic reconstruction has also been extended back by cross-correlating decadal means with the corresponding decadal means of the heliospheric modulation potentials derived from cosmogenic isotope abundances. This has here been done for both linear and third-order fits (solid and dashed lines, respectively) and for both 10Be and 14C cosmogenic isotope records (red and blue, respectively). The model is close to all these empirical extrapolations. Using the polynomial fit shown in Figure 29View Image, the average annual mean FS = 0.7 × 1014Wb for the end of the Maunder minimum modelled by Owens and Lockwood (2012Jump To The Next Citation Point) yields an IMF of B = 2.5 nT. This is quite similar to the floor estimate of Cliver and Ling (2011) of 2.8 nT. Hence, the baselevel CME emergence rate postulated by Owens and Lockwood (2012) comes close to offering a potential explanation of at least the smaller floor estimates. As shown by Figure 35View Image this postulate matches cosmogenic isotope data quite well; however, it remains only a postulate. Note that, in addition, that the cosmogenic isotopes tell us that the Maunder minimum is not the lowest level of solar activity reached in the last 9300 years, and so it remains possible that this minimum still does not set a genuine floor limit to the IMF.

Interestingly, the modelled open solar flux shows cyclic variations during the Maunder minimum. The long time constants of exchange of carbon with the two great terrestrial reservoirs (the biomass and the oceans) means that solar cycle variations cannot be seen in 14C data, but the same is not true for 10Be (Beer et al., 1990). One puzzling observation had been that 10Be continued to show decadal-scale oscillations during the Maunder minimum when no evidence for a magnetic cycle can be found in sunspot data (Beer et al., 1998). Even more puzzling was that, whereas the 10Be abundances at other times are, as expected, in antiphase with sunspots numbers, at the start and end of the Maunder minimum, the 10Be oscillations are in phase with the (small) sunspot cycles (Usoskin et al., 2001). Owens et al. (2012) show how the modelling presented in Figure 35View Image provides the first viable explanation of these two puzzles: at high solar activity levels the solar cycle variation in open solar flux is dominated by the large sunspot cycle variations in the source rate S. On the other hand, at low activity, S becomes small and relatively constant and the open solar flux variations are dominated by the cyclic variations in the loss rate L with the current sheet tilt.


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