The standard equation of magnetic flux transport arises from the radial component of the magnetic induction equation under the assumptions that and .1 These assumptions constrain the radial field component to evolve on a spherical shell of fixed radius, where the time evolution of the radial field component is decoupled from the horizontal field components. Under these assumptions, the evolution of the radial magnetic field, , at the solar surface () is governed by
A wide range of studies have been carried out to determine the best fit profiles for the advection and diffusion processes (see DeVore et al., 1985b,a; Wang et al., 1989b; van Ballegooijen et al., 1998). The parameter study of Baumann et al. (2004) demonstrates the effect of varying many of the model parameters. The most commonly accepted values are the following:
A common treatment of the source term is to include only large magnetic bipoles of flux exceeding 1020 Mx. Extensions of the model to include small-scale magnetic regions are described in Section 2.2.2. While many of the parameters for newly emerging bipoles may be determined observationally, or specified through empirical relationships, the parameter about which there is most disagreement in the literature is the variation of the tilt angle () with latitude, or Joy’s law. While it should in principle be observed directly for each individual magnetic bipole, this is possible only in recent cycles for which magnetogram observations are available. Traditionally the tilt angle was chosen to vary with latitude as , but more recent studies suggest that a much smaller variation with latitude is required (, Schüssler and Baumann, 2006). The tilt angle is a critical quantity as variations can have a significant effect on the net amount of magnetic flux pushed poleward (see Figure 2) and subsequently on the reversal times of the polar fields and amount of open flux. This will be discussed further in Sections 2.2.4 and 4.1.
Since the early flux transport models were produced (DeVore et al., 1985b), new variations have been developed with new formulations and features added by a variety of authors. These include:
The magnetic flux transport models described above have been widely used and extensively compared with observations of the Sun’s large-scale magnetic field. These comparisons are described next for time-scales of up to a single 11-year cycle (Section 2.2.3) and also for multiple solar cycles (Section 2.2.4). In addition, simulations where the profile of meridional flow is changed from that shown in Figure 3b are considered (Section 2.2.5) along with the application of these models to stellar magnetic fields (Section 2.2.6).
It is generally found that for simulations extending up to a full 11-year solar cycle (termed here short term simulations) magnetic flux transport models are highly successful in reproducing the key features of the evolution of the Sun’s radial magnetic field. These include:
A full discussion of these results along with their historical development can be found in the review by Sheeley Jr (2005).
Some short term studies have used magnetic flux transport models to predict the nature of the polar magnetic field. These studies assume that the polar fields are solely due to the passive transport of magnetic flux from low to high latitudes on the Sun. An example is Durrant et al. (2001, 2002) where the authors consider a detailed study of high latitude magnetic plumes. These plumes are produced from poleward surges of magnetic flux that originate from activity complexes (Gaizauskas et al., 1983). While a good agreement was found between observations and models, the authors did find some small discrepancies. They attributed these to small bipole emergences outside the normal range of active latitudes considered in flux transport models. This indicates that the field at high latitudes is not solely the result of magnetic fields transported from low latitudes and a local dynamo action may play a role. To date the only surface flux transport models to include such emergences are those of Schrijver (2001) and Worden and Harvey (2000).
In an additional study, Durrant (2002) and Durrant and Wilson (2003) investigated in detail the reversal times of the polar magnetic fields, comparing the reversal times deduced from KP normal component magnetograms with those found in synoptic flux transport simulations. To test whether magnetic observations or simulations gave the best estimate they deduced the locations of PILs at high latitudes from H filament data. They then compared these to (i) the observed PIL locations deduced from Kitt Peak data and (ii) the simulated PILs from the synoptic flux transport equations. The study found that above 70° latitude the location of the PIL – as given by the H filaments – was in fact better determined by the flux transport process rather than the direct magnetic field observations. This was due to the high level of uncertainty and error in measuring the field at high latitudes as a result of foreshortening and solar B angle effects (Worden and Harvey, 2000). However, the simulation had to be run for over 20 rotations so that the high-latitude fields in the polar regions were solely the product of flux transport processes. This ensured that any systematic errors in the observations at high latitudes were removed.
A recent application of magnetic flux transport models has been to improve forecasts of the solar 10.7 cm (2.8 GHz) radio flux, using an empirical relation between this quantity and total “sunspot” and “plage” fields in the simulation (Henney et al., 2012). The 10.7 cm radio flux is widely used by the space weather community as a proxy for solar activity, with measurements dating back to 1947. For other aspects of space weather, flux transport models need to be coupled to coronal models: these applications will be discussed in Section 4.
Agreement between observations and magnetic flux transport simulations is generally good over time-scales of less than 11 years. But when magnetic flux transport simulations are run for multiple 11-year solar cycles, some inconsistencies are found. One of the first long term simulations was carried out by Schrijver et al. (2002) who simulated the photospheric field over 32 cycles from the Maunder minimum (1649) to 2002. With detailed observations of magnetic bipoles available only for the last few solar cycles, the authors used synthetic data, scaling the activity level to the observed sunspot number. They also assumed that the flux transport parameters remained the same from one cycle to the next, and used a meridional flow profile that peaked at mid-latitudes with a value of 14 m s–1. Since polar field production varies with the amount of emerging flux, the authors found that if the high latitude field is solely described by the passive advection of magnetic flux from the active latitudes, then reversals of the polar field within each cycle may be lost. This is especially true if a series of weak cycles follows stronger ones. Although reversals were lost, they did return in later cycles, but the reversal in Cycle 23 did not match the observed time (see Figure 1 of Schrijver et al., 2002). To ensure that reversals occurred in every cycle, Schrijver et al. (2002) introduced a new exponential decay term for the radial field in Equation (1). This acts to reduce the strength of the large scale field over long periods of time and prevents excess polar fields from building up. They found that a decay time of 5 – 10 yr is optimal in allowing polar field reversals to occur from one cycle to the next. Later, Baumann et al. (2006) also found that a decay term was required in long term simulations to maintain the reversal of the polar field. They formulated the decay term in terms of 3D diffusion of the magnetic field constrained on the 2D solar surface.
While Schrijver et al. (2002) and Baumann et al. (2006) introduced a new physical term into the 2D flux transport model, Wang et al. (2002a) considered a different approach. Rather than keeping the advection due to meridional flow constant from one cycle to the next, they varied the strength of meridional flow such that stronger cycles were given a faster flow. If a factor of two change occurs between strong and weak cycles, then reversals of the polar field may be maintained (see Figure 2 of Wang et al., 2002a). This is because faster meridional flow leads to less flux canceling across the equator, and less net flux transported poleward. If correct, this suggests that in stronger cycles the magnetic fluxes in each hemisphere are more effectively isolated from one another. To complicate matters further, Cameron et al. (2010) put forward a third possibility for maintaining polar field reversals. This was based around an observed cycle-to-cycle variation in the tilt angle of sunspot groups (Dasi-Espuig et al., 2010). If there is an anti-correlation between cycle strength and tilt angles, where stronger cycles are assumed to have smaller tilt angles, then neither variable rates of meridional flow, or extra decay terms are required to maintain polar field reversals from one cycle to the next. To show this, Cameron et al. (2010) consider a simulation extending from 1913 – 1986 and show that with such a tilt angle variation, the polar field reversal may be maintained. Although a decreased tilt angle alone was sufficient to maintain polar field reversals in this study, a longer term study by Jiang et al. (2011b) found that the radial decay term had to be re-introduced when carrying out simulations from 1700 to present (Jiang et al., 2011a). The authors attributed this to inaccuracies in the input data of observed activity levels. However, it is not clear that observed tilt angles in the recent Cycle 23 were sufficiently reduced for this alone to explain the low polar fields in 2008 (Schrijver and Liu, 2008).
This discussion indicates that multiple combinations of model parameters may produce qualitatively the same result. This is illustrated by the study of Schrijver and Liu (2008) who consider the origin of the decreased axial dipole moment found in 2008 compared to 1997. They simulated the global field from 1997 – 2008 and deduced that their previously included decay term was insufficient by itself to account for the lower dipole moment. Instead, they found that to reproduce the observations a steeper meridional flow gradient is required at the equator. This steeper gradient effectively isolates the hemispheres, as in Wang et al. (2002a). However, in contrast to Wang et al. (2002a), in their study Schrijver and Liu (2008) did not have to increase the meridional flow rate, which was maintained at around 10 m s–1. In a similar study, Jiang et al. (2011c) illustrated two additional ways of reproducing the lower polar field strengths in 2008, namely (i) a 28% decrease in bipole tilt angles, or (ii) an increase in the meridional flow rate from 11 m s–1 to 17 m s–1. They, however, found that the first case delayed the reversal time by 1.5 yr, so was inconsistent with observations. Unfortunately, all of these possible solutions are within current observational limits, so none can be ruled out.
While Wang et al. (2002a) and Schrijver and Liu (2008) introduce overall variations of the meridional flow profile and rate, they maintain a basic poleward flow profile in each hemisphere. Recent studies have considered more significant (and controversial) changes to the meridional flow, motivated by helioseismic observations. Two broad types of variation have been considered: a counter-cell near the poles, and an inflow towards activity regions.
Jiang et al. (2009) considered the possibility of the existence of a counter-cell of reversed meridional flow near the poles. This change was motivated directly by observations showing that in Cycles 21 and 22 the polar field was strongly peaked at the poles, while in Cycle 23 it peaked at a latitude of 75° and then reduced by nearly 50% closer to the pole (Raouafi et al., 2008). While the observed structure in Cycles 21 and 22 was consistent with a meridional flow profile that either extended all the way to the poles, or switched off at around 75°, the Cycle 23 structure was not. To investigate what form of meridional flow could reproduce this, Jiang et al. (2009) introduced a counter-cell of meridional flow beyond 70° latitude and showed that the observed distribution of magnetic flux in Cycle 23 could be achieved with a counter-cell rate of 2 – 5 m s–1. The authors showed that the counter-cell need not be constant and that similar results could be obtained if it existed only in the declining phase. To date existence of such a counter-cell has not been verified beyond doubt even though some results from helioseismology support its existence (Haber et al., 2002; González Hernández et al., 2006). Also, due to the difference in profile of polar magnetic fields in Cycles 21 and 22 it is unlikely that such a counter-cell existed in these cycles.
The second type of flow perturbation suggested by observations are activity-dependent inflows towards the central latitude of emergence of the butterfly diagram (Gizon and Rempel, 2008; González Hernández et al., 2008). At least in part, this modulation of the axisymmetric meridional flow component results from the cumulative effect of the sub-surface horizontal flows converging towards active regions (Hindman et al., 2004). In an initial study, DeRosa and Schrijver (2006) simulated the effect of (non-axisymmetric) inflows towards active regions by adding an advection term to the flux transport model, its velocity proportional to the horizontal gradient of radial magnetic field. They found that if surface speeds exceed 10 m s–1 then an unrealistic “clumping” of magnetic flux that is not observed occurs. More recently, Jiang et al. (2010b) have considered the effect of an axisymmetric flow perturbation towards active latitudes of speed 3 – 5 m s–1, similar to perturbations observed during Cycle 23. The authors consider how such a flow effects the polar field distribution though simulated solar cycles. The main effect is to decrease the separation of magnetic bipoles, subsequently leading to more cancellation in each hemisphere and less net flux transport poleward. This leads to an 18% decrease in the polar field strength compared to simulations without it. While this is a significant decrease, the authors note that it cannot (alone) account for the weak polar fields observed at the end of Cycle 23 as those fields decreased by more than a factor of 2. However, in conjunction with a variable meridional flow from one cycle to the next (Wang et al., 2002a) or the steepening gradient of the flow (Schrijver and Liu, 2008), the inflow may have a significant effect. Cameron and Schüssler (2010) show that one can incorporate such axisymmetric flow perturbations by setting the speed at a given time proportional to the latitudinal gradient of longitude-averaged . Not only does this generate appropriate inflows, but these can explain the observed solar cycle variation of the Legendre component of meridional flow (Hathaway and Rightmire, 2010) without changing the background flow speed.
Due to advances in measuring stellar magnetic fields, magnetic flux transport models have recently been applied in the stellar context. One of their main applications has been to consider how polar or high latitude intense field regions or stellar spots may arise in rapidly rotating stellar systems (Strassmeier and Rice, 2001). Initial studies (Schrijver and Title, 2001; Mackay et al., 2004) used as a starting point parameters from solar magnetic flux transport simulations. These parameters were then varied to reproduce the key observational properties of the radial magnetic fields on stars as deduced through ZDI measurements. In the paper of Schrijver and Title (2001) the authors consider a very active cool star of period 6 days. A key feature of their simulations is that they fix all parameters to values determined for the Sun and only vary the emergence rate to be 30 times solar values. They show that even if solar emergence latitudes are maintained, then the flux transport effects of meridional flow and surface diffusion are sufficient to transport enough flux to the poles to produce a polar spot. In Figure 5, a comparison of typical solar (Figure 5a) and stellar (Figure 5b) magnetic field configurations from Figure 2 of Schrijver and Title (2001) can be seen. For the case of the Sun the magnetic fields saturate at ± 70 Mx cm–2 and for the star ± 700 Mx cm–2. Both images show the polar region from a latitude of 40°. A clear difference can be seen, where for the Sun the pole has a weak unipolar field. For the rapidly rotating star there is a unipolar spot with a ring of strong opposite polarity flux around it. The existence of this ring is partly due to the non-linear surface diffusion used in Schrijver and Title (2001) which causes intense field regions to diffuse more slowly (see Section 2.2.2).
In contrast to the unipolar poles modeled by Schrijver and Title (2001), some ZDI observations of rapidly rotating stars show intermingling of opposite polarities at high latitudes, where intense fields of both polarities lie at the same latitude and are not nested. An example of this can be seen in AB Dor (Figure 1b) which has a rotation period of 1/2 day. Mackay et al. (2004) showed that in order to produce strong intermingled polarities within the polar regions more significant changes are required relative to the solar flux transport model. These include increasing the emergence latitude of new bipoles from 40° to 70° and increasing the rate of meridional flow from 11 m s–1 to 100 m s–1. Increased emergence latitudes are consistent with an enhanced Coriolis force deflecting more flux poleward as it travels through the convection zone (Schüssler and Solanki, 1992). However, at the present time, the predicted enhanced meridional flow is still below the level of detection. While the simulations of Schrijver and Title (2001) and Mackay et al. (2004) have successfully reproduced key features in the magnetic field distributions of rapidly rotating stars, as yet they do not directly simulate the observations as has been done for the Sun. Presently, there is insufficient input data on the emergence latitudes of new bipoles in order to carry out such simulations. In the study of Işık et al. (2007) the authors used magnetic flux transport simulations to estimate the lifetime of starspots as they are transported across the surface of a star. The authors show that many factors may effect the star spot lifetime, such as the latitude of emergence and differential rotation rate. In particular the authors show that for rapidly rotating stars the lifetime of spots may be 1 – 2 months, however the lifetime is lower for stars with strong differential rotation (AB Dor) compared to those with weak differential rotation (HR 1099).
In recent years magnetic flux transport simulations have been combined with models for the generation and transport of magnetic flux in the stellar interior (Holzwarth et al., 2006; Işık et al., 2011) to predict interior properties on other stars. The first study to link the pre- and post-eruptive transport properties was carried out by Holzwarth et al. (2006). In this study, the authors quantified what effect the enhanced meridional flow predicted by Mackay et al. (2004) would have on the dynamics and displacement of magnetic flux tubes as they rise though the convective zone (Moreno-Insertis, 1986; Schüssler et al., 1994). The authors found that the enhanced meridional flow leads to a non-linear displacement of the flux tubes, where bipoles were displaced to emerge at higher latitudes. This was consistent with the required higher latitude of emergence to produce the desired intermingling in the poles. However, in a more detailed model, Işık et al. (2011) combine a thin-layer - dynamo model with a convective zone transport model and surface flux transport model to provide a complete description of the evolution of magnetic field on stars. Through this combined model the authors show that for rapid rotators (period 2 days), due to the effect of the Coriolis force, the surface signature of the emergence and subsequent transport may not be a clear signal of the dynamo action that created it. This has important consequences for both the observation and interpretation of magnetic fields on other stars.
Living Rev. Solar Phys. 9, (2012), 6
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