2.2 Magnetic flux transport simulations

On large spatial scales, once new magnetic flux has emerged on the Sun, it evolves through the advection processes of differential rotation (Snodgrass, 1983Jump To The Next Citation Point) and meridional flow (Duvall Jr, 1979; Hathaway and Rightmire, 2010Jump To The Next Citation Point). In addition, small convective cells such as super-granulation lead to a random walk of magnetic elements across the solar surface. On spatial scales much larger than super-granules this random walk may be modeled as a diffusive process (Leighton, 1964Jump To The Next Citation Point). Magnetic flux transport simulations (Sheeley Jr, 2005Jump To The Next Citation Point) apply these effects to model the large-scale, long-time evolution of the radial magnetic field B (πœƒ,Ο•, t) r across the solar surface. In Section 2.2.1 the basic formulation of these models is described. In Section 2.2.2 extensions to the standard model are discussed and, finally, in Sections 2.2.32.2.6 applications of magnetic flux transport models are considered.

2.2.1 Standard model

The standard equation of magnetic flux transport arises from the radial component of the magnetic induction equation under the assumptions that vr = 0 and ∂ βˆ•∂r = 0.1 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, B r, at the solar surface (R = 1 βŠ™) is governed by

( ( ) ) 2 ∂Br- -1---∂-- ∂Br- ∂Br- --D---∂-Br- ∂t = sin πœƒ ∂πœƒ sinπœƒ − u (πœƒ )Br + D ∂πœƒ − Ω (πœƒ)∂ Ο• + sin2 πœƒ ∂Ο•2 + S (πœƒ,Ο•,t), (1 )
where Ω(πœƒ) and u(πœƒ) represent the surface flows of differential rotation and meridional flow, respectively, which passively advect the field, D is the isotropic diffusion coefficient representing superganular diffusion, and finally S (πœƒ,Ο•,t) is an additional source term added to represent the emergence of new magnetic flux. Figure 2View Image illustrates two numerical solutions to the flux transport equation when S = 0. Both are initialised with a single bipole in the northern hemisphere, which is then evolved forward in time for 30 solar rotations. The simulations differ only in the tilt of the initial bipole: in the left column, the bipole satisfies Joy’s law, while in the right column both polarities lie at the same latitude. The effect of Joy’s law has a significant impact on the strength and distribution of Br across the surface of the Sun and, in particular, in the polar regions.

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., 1985bJump To The Next Citation Point,a; Wang et al., 1989bJump To The Next Citation Point; van Ballegooijen et al., 1998Jump To The Next Citation Point). 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:

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Figure 2: Evolution of the radial component of the magnetic field (Br) at the solar surface for a single bipole in the northern hemisphere with (a) – (c) initial tilt angle of γ = 20° and (d) – (f) γ = 0°. The surface distributions are shown for (a) and (d) the initial distribution, (b) and (e) after 15 rotations, and (c) and (f) after 30 rotations. White represents positive flux and black negative flux and the thin solid line is the Polarity Inversion Line. The saturation levels for the field are set to 100 G, 10 G, and 5 G after 0, 15, and 30 rotations, respectively.
  1. Differential Rotation: The form that best agrees with the evolution of magnetic flux seen on the Sun (DeVore et al., 1985bJump To The Next Citation Point) is that of Snodgrass (1983Jump To The Next Citation Point). The profile (Figure 3View Imagea) is given by
    2 4 Ω (πœƒ) = 13.38 − 2.30 cos πœƒ − 1.62cos πœƒ deg βˆ•day, (2)
    and was determined by cross-correlation of magnetic features seen on daily Mt. Wilson magnetogram observations. The key effect of differential rotation is to shear magnetic fields in an east-west direction, where the strongest shear occurs at mid-latitudes (see Figure 3View Imagea). This produces bands of alternating positive and negative polarity as you move poleward (visible after 15 rotations in Figure 2View Image). These bands result in steep meridional gradients which accelerate the decay of the non-axisymmetric2 field during periods of low magnetic activity (DeVore, 1987Jump To The Next Citation Point). On the Sun the timescale for differential rotation to act is τdr = 2πβˆ•(Ω (0) − Ω (90)) ∼ 1βˆ•4 year. Within magnetic flux transport simulations the profile (2View Equation) is either applied directly, simulating the actual rotation of the Sun, or with the Carrington rotation rate (13.2 deg/day) subtracted (van Ballegooijen et al., 1998Jump To The Next Citation Point; McCloughan and Durrant, 2002Jump To The Next Citation Point).
    View Image

    Figure 3: (a) Profile of differential rotation Ω versus latitude (Snodgrass, 1983) that is most commonly used in magnetic flux transport simulations. (b) Profiles of meridional flow u versus latitude used in various studies. The profiles are from Schüssler and Baumann (2006Jump To The Next Citation Point) (solid line), van Ballegooijen et al. (1998Jump To The Next Citation Point) (dashed line), Schrijver (2001Jump To The Next Citation Point) (dotted line), and Wang et al. (2002bJump To The Next Citation Point) (dash-dot and dash-dot-dot-dot lines). This figure is based on Figure 3 of Schüssler and Baumann (2006Jump To The Next Citation Point).
  2. Supergranular Diffusion: This represents the effect on the Sun’s large-scale magnetic field of the small-scale convective motions of supergranules. The term was first introduced by Leighton (1964Jump To The Next Citation Point) to represent the effect of the non-stationary pattern of supergranular cells dispersing magnetic flux across the solar surface. In addition it describes the cancellation of magnetic flux (see Figure 2View Image) when positive and negative magnetic elements encounter one-another. Initial estimates based on obtaining the correct reversal time of the polar fields (without including meridional flow) were of a diffusion coefficient of 2 −1 D ∼ 770 –1540 km s. However, once meridional flow was included to aid the transport of magnetic flux poleward, the value was lowered to around 200 –600 km2 s−1 which better agrees with observational estimates and is commonly used in models today (DeVore et al., 1985bJump To The Next Citation Point; Wang et al., 1989bJump To The Next Citation Point). Globally this gives a time-scale τ = R2 βˆ•D ∼ 34 –80 yr mf βŠ™, however when considering non-global length scales such as that of individual active regions the timescale is much shorter. Some models have introduced a discrete random walk process as an alternative to the diffusion term which will be discussed in Section 2.2.2 (e.g., Wang and Sheeley Jr, 1994; Schrijver, 2001Jump To The Next Citation Point).
  3. Meridional Flow: This effect was the last to be added to what is now known as the standard magnetic flux transport model. It represents an observed weak flow that pushes magnetic flux from the equator to the poles in each hemisphere. The exact rate and profile applied varies from author to author, but peak values of 10 – 20 m s–1 are commonly used. Figure 3View Imageb shows a number of meridional flow profiles that have been used by different authors. The typical time-scale for meridional flow is τmf = R βŠ™βˆ•u(πœƒ) ∼ 1 –2 yr. As current measurements of meridional flow are at the limits of detection, for both the flow rate and profile, this has lead to some variations in the exact profile and values used which will be discussed in more detail in Section 2.2.4. The first systematic study of the effects of meridional flow on the photospheric field was carried out by DeVore et al. (1984). This showed that to obtain a realistic distribution and strength for the polar field, the meridional flow profile must peak at low to mid-latitudes and rapidly decrease to zero at high latitudes. The inclusion of meridional flow was a key development which meant that much lower rates of the diffusion coefficient could be applied, while still allowing the polar fields to reverse at the correct time. It also aids in producing the observed “topknot” latitudinal profile of the polar field (more concentrated than a dipole, Sheeley Jr et al., 1989), and in reproducing the strong poleward surges observed in the butterfly diagram (Wang et al., 1989a). In recent years more significant changes to the profile of meridional flow have been suggested. This will be discussed in Section 2.2.4 and Section 2.2.5.
  4. Magnetic Flux Emergence: The final term in Equation (1View Equation) is a time-dependent source term which represents a contribution to the radial magnetic field from the emergence of new magnetic bipoles. Most flux transport simulations carry out emergence in a semi-empirical way where the emergence is carried out instantaneously, so that growth of the new bipole is not considered. Instead, only its decay under the action of the advection and diffusion process described above are followed. Inclusion of this term is critical in simulations extending over more than one rotation, to ensure that accuracy of the photospheric field is maintained. Within the literature the source term has been specified in a number of ways and reproduces the main properties of the butterfly diagram, along with varying levels of magnetic activity through single cycles and from one cycle to the next. Different ways in which the source term has been specified include:
    1. Observationally determining the properties of new bipoles from daily or synoptic magnetograms so that actual magnetic field configurations found on the Sun may be reproduced (Sheeley Jr et al., 1985Jump To The Next Citation Point; Yeates et al., 2007Jump To The Next Citation Point). Statistical variations of these properties have been applied to model multiple solar cycles of varying activity.
    2. Producing synthetic data sets from power law distributions (Harvey and Zwaan, 1993; Schrijver and Harvey, 1994) where the power laws specify the number of bipoles emerging at a given time with a specific area or flux (van Ballegooijen et al., 1998Jump To The Next Citation Point; Schrijver, 2001Jump To The Next Citation Point).
    3. Using observations of sunspot group numbers to specify the number of bipoles emerging (Schrijver et al., 2002Jump To The Next Citation Point; Baumann et al., 2006Jump To The Next Citation Point; Jiang et al., 2010aJump To The Next Citation Point) where the flux within the bipoles can be specified through empirical sunspot area-flux relationships (Baumann et al., 2006Jump To The Next Citation Point; Jiang et al., 2010a). Recently Jiang et al. (2011aJump To The Next Citation Point) extended this technique back to 1700 using both Group (Hoyt and Schatten, 1998) and Wolf (Wolf, 1861) sunspot numbers.
    4. Assimilating observed magnetograms directly into the flux transport simulations (Worden and Harvey, 2000Jump To The Next Citation Point; Schrijver et al., 2003Jump To The Next Citation Point; Durrant et al., 2004Jump To The Next Citation Point). Worden and Harvey (2000Jump To The Next Citation Point) use the flux transport model to produce evolving magnetic synoptic maps from NSO Kitt Peak data. They develop a technique where full-disk magnetograms are assimilated when available and a flux transport model is used to fill in for unobserved or poorly observed regions (such as the far side of the Sun or the poles). Schrijver et al. (2003Jump To The Next Citation Point) use a similar technique but with full-disk MDI observations, demonstrating it over a 5 yr period. For the near side of the Sun, the MDI observations are inserted every 6 h to reproduce the actual field over a 60° degree window where the measurements are most accurate. The authors show that the magnetic flux transport process correctly predicts the return of magnetic elements from the far side, except for the case of flux that emerged on the far-side. To account for this, the authors also include far side acoustic observations for the emergence of new regions (Lindsey and Braun, 2000; Braun and Lindsey, 2001). In contrast to the method of Schrijver et al. (2003Jump To The Next Citation Point), which inputs observations from daily MDI magnetograms, Durrant et al. (2004) inserted fields from synoptic magnetograms once per solar rotation for all latitudes between ± 60°. They used this to investigate the transport of flux poleward and the reversal of the polar field. Recently, the Worden and Harvey (2000Jump To The Next Citation Point) model has been incorporated in a more rigorous data assimilation framework to form the Air Force Data Assimilative Photospheric Flux Transport Model (ADAPT, Arge et al., 2010; Henney et al., 2012Jump To The Next Citation Point). An ensemble of model realisations with different parameter values allow both data and model uncertainties to be incorporated in predictions of photospheric evolution.

    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 (γ) 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 λ as γ ∼ λ βˆ•2, but more recent studies suggest that a much smaller variation with latitude is required (γ ∼ 0.15λ, Schüssler and Baumann, 2006Jump To The Next Citation Point). 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 2View Image) 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.

2.2.2 Extensions

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:

  1. Extending the basic flux transport model for the large-scale field to include the emergence of small-scale fields down to the size of ephemeral regions (Worden and Harvey, 2000Jump To The Next Citation Point; Schrijver, 2001Jump To The Next Citation Point). Such small scale fields are a necessary element to model the magnetic network and simulate chromospheric radiative losses. Including the small scale emergences leads to a much more realistic description of the photospheric field where discrete magnetic elements can be seen at all latitudes (compare Figure 4View Image to Figure 2View Image), however since they are randomly oriented small-scale regions do not have a significant effect on large-scale diffusion or on the polar field (Wang and Sheeley Jr, 1991; Worden and Harvey, 2000Jump To The Next Citation Point). To resolve such small-scales, Schrijver (2001Jump To The Next Citation Point) introduced a particle-tracking concept along with rules for the interaction of magnetic elements with one-another. Another important extension introduced by Schrijver (2001Jump To The Next Citation Point) was magneto-convective coupling. With its introduction the early decay of active regions could be considered, where strong field regions diffuse more slowly due to the suppression of convection.
  2. Introducing an additional decay term into the basic flux transport model (Schrijver et al., 2002Jump To The Next Citation Point; Baumann et al., 2006Jump To The Next Citation Point). In the paper of Baumann et al. (2006Jump To The Next Citation Point) this term is specified as a linear decay of the mode amplitudes of the radial magnetic field. The term approximates how a radial structure to the magnetic field, along with volume diffusion, would affect the radial component that is constrained to lie on a spherical shell. The authors only include a decay term based on the lowest order radial mode, as it is the only radial mode with a sufficiently long decay time to affect the global field. Through simulating the surface and polar fields from Cycles 13 – 23 (1874 – 2005) and requiring the simulated polar fields to reverse at the correct time as given by both magnetic and proxy observations, the authors deduce a volume diffusivity of the order of 2 −1 50– 100 km s.
  3. Coupling the magnetic flux transport model to a coronal evolution model so that both the photospheric and coronal magnetic fields evolve together in time. The first study to consider this was carried out by van Ballegooijen et al. (1998Jump To The Next Citation Point). Limitations of this initial model were that in the coronal volume the radial (Br) and horizontal (Bπœƒ,B Ο•) field components evolve independently from one another, and that no force balance was considered for the coronal field. Later developments (van Ballegooijen et al., 1998Jump To The Next Citation Point, 2000Jump To The Next Citation Point; Mackay and van Ballegooijen, 2006Jump To The Next Citation Point, see Section 3.2.3) removed all of these restrictive assumptions so that a fully coupled evolution of the coronal field along with radial diffusion, radial velocities, and force balance occurred. In an additional study the technique was also extended to include a simplified treatment of the convective zone (van Ballegooijen and Mackay, 2007).
  4. Reformulating the flux transport equation into a “synoptic” transport equation, that evolves synoptic magnetic field maps such as those produced by NSO/Kitt Peak3 from one Carrington rotation to the next (McCloughan and Durrant, 2002). Their synoptic transport equation takes the form
    ( ) 2 ∂ℬr- f(πœƒ)-∂-- vΟ•(πœƒ)f-(πœƒ)-∂ℬr- Df-(πœƒ)-∂- ∂-ℬr Df-(πœƒ)-∂-ℬr- ∂τ = − sin πœƒ ∂πœƒ (sin πœƒu(πœƒ)ℬr) − sin πœƒ ∂ Ο• + sin πœƒ ∂ πœƒ sin πœƒ ∂πœƒ + sin2 πœƒ ∂Ο•2 , (3)
    where ℬr represents the synoptic magnetogram, τ a time index such that integer values of 2πβˆ•Ω correspond to successive synoptic magnetograms, and f(πœƒ) = 1βˆ•(1 + vΟ•(πœƒ)βˆ•(Ω sinπœƒ)). The individual terms in Equation (3View Equation) are similar to those of Equation (1View Equation) but modified to take into account that, due to differential rotation, different latitudes return to central meridian at different times.
  5. Although the surface flux transport model is primarily considered in 2D, Leighton (1964) also introduced a reduced 1-D form where the large-scale radial field is spatially averaged in the azimuthal direction. Due to the form of Equation (2View Equation), the differential rotation Ω (πœƒ) plays no role in such a model. Recent applications of this 1-D model have been to reconcile surface flux transport with results from axisymmetric, kinematic dynamo simulations in the r-πœƒ plane. Cameron and Schüssler (2007), through describing the emergence of magnetic flux based on sunspot records in a similar manner to that used by Dikpati et al. (2006) and Dikpati and Gilman (2006), determine whether the 1-D surface flux transport model has any predictive properties. They show that in some cases a significant positive correlation can be found between the amount of flux canceling across the equator and the strength of the next cycle. However, when detailed observations of bipole emergences are added, they find that any predictive power may be lost. Cameron et al. (2012) have used the same model to derive constraints on parameter values in the dynamo models, based on observed flux evolution at the surface.
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Figure 4: Example of the magnetic flux transport simulations of Schrijver (2001Jump To The Next Citation Point) which apply a particle-tracking concept to simulate global magnetic fields down to the scale of ephemeral regions. Image reproduced by permission from Schrijver (2001Jump To The Next Citation Point), copyright by AAS.

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 3View Imageb are considered (Section 2.2.5) along with the application of these models to stellar magnetic fields (Section 2.2.6).

2.2.3 Short term applications of magnetic flux transport models

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:

  1. The 27 day and 28 – 29 day recurrent patterns seen in plots of the Sun’s mean line-of-sight field. These patterns can be respectively attributed to differential rotation and the combined effect of differential rotation with magnetic flux emergence (Sheeley Jr et al., 1985).
  2. Reproducing the rigid rotation of large-scale magnetic features at the equatorial rate, caused by a balance between differential rotation and a poleward drift due to the combined effects of meridional flow and surface diffusion (DeVore, 1987; Sheeley Jr et al., 1987; Wang et al., 1989bJump To The Next Citation Point).
  3. Reproducing the reversal time of the polar fields along with the strength and top-knot profile (DeVore and Sheeley Jr, 1987; Durrant and Wilson, 2003Jump To The Next Citation Point).
  4. Matching qualitatively the strength and distribution of the radial magnetic field at the photosphere. Examples include the formation of switchbacks of polarity inversions lines at mid-latitudes due to the dispersal of magnetic flux from low latitude active regions (Wang et al., 1989b; van Ballegooijen et al., 1998Jump To The Next Citation Point) and the return of magnetic elements from the far-side of the Sun (Schrijver and DeRosa, 2003Jump To The Next Citation Point).

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 (2000Jump To The Next Citation Point).

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 Hα 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 Hα 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.

2.2.4 Multiple solar cycle applications of magnetic flux transport models

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. (2002Jump To The Next Citation Point) 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., 2002Jump To The Next Citation Point). To ensure that reversals occurred in every cycle, Schrijver et al. (2002Jump To The Next Citation Point) introduced a new exponential decay term for the radial field in Equation (1View Equation). 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. (2006Jump To The Next Citation Point) 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. (2002aJump To The Next Citation Point) 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., 2002aJump To The Next Citation Point). 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. (2010Jump To The Next Citation Point) 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. (2010Jump To The Next Citation Point) 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, 2008Jump To The Next Citation Point).

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 (2008Jump To The Next Citation Point) 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. (2002aJump To The Next Citation Point). However, in contrast to Wang et al. (2002aJump To The Next Citation Point), in their study Schrijver and Liu (2008Jump To The Next Citation Point) 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.

2.2.5 Variations in the meridional flow profile

While Wang et al. (2002aJump To The Next Citation Point) and Schrijver and Liu (2008Jump To The Next Citation Point) 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.

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Figure 5: Comparison of (a) solar and (b) stellar magnetic field configurations from Schrijver and Title (2001Jump To The Next Citation Point). For the stellar system all flux transport parameters are held fixed to solar values and only the emergence rate is increased to 30 times solar values. For the case of the Sun the magnetic fields saturate at ± 70 Mx cm–2 and for the star ± 700 Mx cm–2. Image reproduced by permission from Schrijver and Title (2001Jump To The Next Citation Point), copyright by AAS.

Jiang et al. (2009Jump To The Next Citation Point) 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 Br. Not only does this generate appropriate inflows, but these can explain the observed solar cycle variation of the P21 Legendre component of meridional flow (Hathaway and Rightmire, 2010) without changing the background flow speed.

2.2.6 Stellar applications of magnetic flux transport models

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, 2001Jump To The Next Citation Point; Mackay et al., 2004Jump To The Next Citation Point) 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 (2001Jump To The Next Citation Point) 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 5View Image, a comparison of typical solar (Figure 5View Imagea) and stellar (Figure 5View Imageb) magnetic field configurations from Figure 2 of Schrijver and Title (2001Jump To The Next Citation Point) 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 (2001Jump To The Next Citation Point) 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 (2001Jump To The Next Citation Point), 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 1View Imageb) which has a rotation period of 1/2 day. Mackay et al. (2004Jump To The Next Citation Point) 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. (2004Jump To The Next Citation Point) 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., 2006Jump To The Next Citation Point; Işık et al., 2011Jump To The Next Citation Point) 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 α-Ω 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.


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