Centennial variations in sunspot number, open solar flux and streamer belt width: 3. Modeling

From the variation of near‐Earth interplanetary conditions, reconstructed for the mid‐19th century to the present day using historic geomagnetic activity observations, Lockwood and Owens (2014) have suggested that Earth remains within a broadened streamer belt during solar cycles when the Open Solar Flux (OSF) is low. From this they propose that the Earth was immersed in almost constant slow solar wind during the Maunder minimum (c. 1650–1710). In this paper, we extend continuity modeling of the OSF to predict the streamer belt width using both group sunspot numbers and corrected international sunspot numbers to quantify the emergence rate of new OSF. The results support the idea that the solar wind at Earth was persistently slow during the Maunder minimum because the streamer belt was broad.


Reconstructions of Open Solar Flux from Historic Geomagnetic Activity Data
The (signed) open solar flux (OSF) F S was first reconstructed by Lockwood et al. [1999] from geomagnetic data for 1868 onward using the aa activity index and its 27 day recurrence index. Svalgaard and Cliver [2007] noted that both the near-Earth interplanetary magnetic field (IMF), B, and solar wind speed, V SW , could both be reconstructed using pairs of indices that have different dependencies on these near-Earth interplanetary parameters. This was exploited by Rouillard et al. [2007], who used the separate B and V SW estimates with Parker spiral theory (and a statistical allowance for excess flux effect, discussed below in section 1.4) to reconstruct the open solar flux variation. The development of these reconstructions has been reviewed by Lockwood [2013].
A recent analysis has been presented in a series of four papers by Lockwood et al. [2013aLockwood et al. [ , 2013bLockwood et al. [ , 2014cLockwood et al. [ , 2014d which used four pairings of magnetic indices to reconstruct B, V SW , and F S from 1845 to the present day with full analysis of the uncertainties in each case. These uncertainties were computed at the 2σ level using a Monte-Carlo fitting technique and allow for regression fit errors, the IMF orientation factor (relevant because geomagnetic activity responds to the southward component of the IMF), in situ satellite measurement errors, geomagnetic instrument intercalibration errors, geomagnetic observatory calibration drift, and site changes. The resulting variations are shown in Figure 1 (from Lockwood et al. [2014d]). The grey areas mark the ±2σ uncertainty band in each case, and these uncertainties are largest for the earliest data (because of geomagnetic observation uncertainties) but are remarkably small for all years. Note that the annual means of the in situ data (the blue dots) sometimes lie outside the reconstruction uncertainty band because of the effect of data gaps in the satellite measurements: these are not a factor in the reconstructions which are based on correlations between annual means with piecewise removal of geomagnetic data at times corresponding to satellite data gaps . Lockwood et al. [2014d] show that the reconstruction of F S using the method of Lockwood et al. [1999] also lies within the uncertainty band in Figure 1f.
Thus, we can reconstruct these three parameters with a high degree of certainty back to 1845, just 13 years after the establishment by Gauss of the first magnetic observatory in Göttingen. It is important to make a clear distinction between OSF which, like sunspot number, is an indicator of the global state of the Sun, and B and V SW which are samples of the heliosphere at a single point in the ecliptic plane.
distinction between "open" solar flux as being field lines that extend beyond the edge of the schematic and "closed" field lines that form closed loops within the schematic. Such schematics often lead to the idea that open flux that only has one "foot point" still attached to the Sun instead of two: however, this is a false definition because, unless one invokes magnetic monopoles, all field lines that are still connected to the Sun have two such foot points, however distended the loop between them. Hence, a more precise definition of open flux is that the loop between the two foot points threads a certain surface in the heliosphere. This could either be a real but movable (and non-spherical) boundary (such as the heliopause, the heliospheric termination shock, or the Alfvén surface where the solar wind flow becomes super-Alfvénic) or it could be a nominal boundary, such as a fixed coronal source surface. A less precise definition is that the loop between the two foot points is so distended that one cannot tell where it maps to. In theory, unidirectional or bidirectional heat flux or "strahl" electron distribution functions could distinguish between open and closed flux by this latter definition; however, in practice electron scattering by heliospheric structure into other populations such as "halo" makes even this imprecise distinction ambiguous [Larson et al., 1997;Fitzenreiter et al., 1998;Owens et al., 2008a]. However, these definitions are all academic when it comes to quantifying the total open solar flux, because there is no way of reliably mapping all the field lines (to see if they thread a certain heliospheric surface) or of remotely sensing the electron flows along them (to see if they are unidirectional or bidirectional) as a way of empirically quantifying the total open flux.
The only definition of OSF that gives a measurable quantity is that it is the magnetic flux threading the top of the solar corona. Usually, the "coronal source surface" is employed, taken to be a heliocentric sphere of radius r = 2.5 R ⊙ , where R ⊙ is a mean solar radius. OSF defined this way can be measured in two ways: (1) using "Potential Field Source Surface" (PFSS) modeling of the solar corona based on photospheric  Figures 1b, 1d, and 1f, the agreement between the four reconstructions is so close that some lines cannot be seen as they lie beneath others. Figures 1c, 1e, and 1g show the probability distribution functions (pdfs) from the in situ spacecraft data on B, V SW , and F S , respectively, for 1966-2012 (inclusive) for hourly means (in gray) and annual means (in blue). Note that the pdf scale has been halved for annual data for clarity. [From Lockwood et al., 2014d].
magnetograms and several assumptions [see review by Mackay and Yeates, 2012]; or (2) from in situ measurements or estimates of the radial field in the heliosphere (for example near Earth) [see review by Lockwood, 2013]. However, in the second case, there are some important caveats and corrections caused by effects taking place between the source surface and the observation point which are discussed in section 1.4.
Because it renders it quantifiable, we here adopt the definition that the (signed) OSF is the total flux of one polarity that threads a fixed coronal source surface at r = 2.5 R ⊙ and consider the implications of that definition. So-called "interchange reconnections" [Crooker et al., 2002] [Fisk et al., 1999;Fisk and Schwadron, 2001]) take place simultaneously with the cycles of OSF growth and loss (as the balance between OSF emergence and disconnection changes).

Modeling of the Open Solar Flux Variation
The concept used to model the OSF variation was first introduced by Solanki et al. [2000]. These authors were not concerned with the spatial distribution of open flux on the Sun but applied a global continuity equation to all open flux. They used sunspot number as a basis of their quantification of the emergence rate of open flux and an assumed linear loss rate with a time constant that was fitted to match the OSF reconstruction of Lockwood et al. [1999]. The open flux continuity equation has subsequently been used by many authors with refinements to the production and loss rate formulations used [Lockwood, 2003;Crooker, 2006, 2007;Vieira and Solanki, 2010;Schwadron et al., 2010;Goelzer et al., 2013].
The main difference in the modeling by Lockwood [2003] was only that instead of running the continuity equation forward in time from an assumed open flux of zero at the end of the Maunder minimum (as has been done by Solanki et al.), it was run backward in time starting from the observed open flux at the minimum between solar cycles 22 and 23: this yielded an estimate of the total open flux at the end of the Maunder minimum that was roughly a quarter of the average of space-age values. Owens and Crooker [2006] introduced the use of coronal mass ejection (CME) occurrence frequency and flux content into the computation of the emergence rate, which has subsequently been used by Schwadron et al. [2010], , and Goelzer et al. [2013] and is also employed here. The other differences between the models relate to the computation of the loss rate. Vieira and Solanki [2010] divided open flux into a shortlived and a long-lived component, each with its characteristic loss timescale. Schwadron et al. [2010] subdivided the open solar flux into a CME-associated part and the remainder and derived continuity equations for both parts including the effect of interchange reconnections on both. As discussed in section 1.2, for the measurable definition of OSF used here, this does not change the overall continuity equation so essentially these authors have refined both the production and loss terms. However, this has been achieved at the expense of an increase in the number of free variables. One criticism of the original Solanki et al. [2000] modeling that was often raised was that, in quantifying the emergence and loss rates, there are several parameters that are either free or poorly defined to the extent that fits to observations or reconstructions are not significant. Subdividing the production and loss terms, as done by Vieira and Solanki and Schwadron et al. gives more degrees of freedom, thereby making the fits achieved less statistically significant.
For this reason, the modeling by  was importantly different. These authors argued that new OSF injection is predominantly in the form of CMEs and so used CME fluxes and occurrence rates as a function of sunspot number, as introduced by Owens and Crooker [2006]. Using the OSF variation reconstructed from geomagnetic activity these authors showed that this production term requires a fractional OSF loss rate that depends only on the phase of the solar cycle and is approximately the same in each solar cycle (i.e., it is independent of the cycles' amplitude in sunspot numbers or in other solar activity indices). Furthermore, these authors showed that the solar cycle variation in this fractional OSF loss was very similar to the variation of the Heliospheric Current Sheet tilt-as had been predicted theoretically by Owens et al. [2011b] by considering how current sheet tilt facilitates OSF loss by disconnection. By making the assumption that the emergence rate in the long, low minimum between solar cycles 23 and 24 fell to its Maunder minimum value,  prescribed all inputs to the model and so had removed free fit parameters except one, which was the scaling factor between fractional OSF loss rate and the current sheet tilt.

The Relationship Between Open Solar Flux and the Near-Earth IMF
The continuity models described in section 1.2 predict the OSF and so to compare with observations and reconstructions of the near-Earth interplanetary magnetic field (IMF, B), the relationship between B and OSF must be known and understood. Part of this relationship is given by the theory of the Parker spiral in the heliospheric field [Parker, 1958]. However, there is a second factor that must be recognized and understood, namely the role of heliospheric field lines that do not follow the Parker spiral configuration, such as "folded" and "ortho-gardenhose" flux. This section deals with this second factor.
The Ulysses satellite showed that the average of the radial component of the heliospheric field (B r ) was independent of latitude. How the averaging is done is clearly important (for example if 27 day periods are used, "away" (from the Sun) and "toward" flux will largely cancel in the streamer belt giving a small residual value, whereas they would not cancel in a unipolar polar coronal hole. Smith and Balogh [1995] found latitudinal constancy by separately averaging over toward and away interplanetary sectors in the streamer belt, whereas Lockwood et al. [2004] found that it also applied to the average of the modulus of the radial field, |B r |. Smith [2011] argues that the latter introduces an artefact and that only the sector averaging method should be used: however, as pointed out by Lockwood and Owens [2013], this does not answer the question of how sectors should be defined: for example, Lockwood and Owens showed that using the polarity of B r observed at the spacecraft to define the sectors is mathematically identical to taking the modulus. Hence, the sector-average method depends entirely on the accurate identification of the sector boundaries (at the source surface, and not at the spacecraft), and there will be unknown errors associated with inaccurate definition of these boundaries.
The latitudinal constancy of |B r | found by Lockwood et al. [2004] means that averaging over a solar rotation allows us to quantify the signed flux threading a heliocentric sphere of radius r as F(r) = 4πr 2 < |B r |>/2. Applying this to r = 2.5 R ⊙ gives, by the definition we employ, the OSF, F S = F(r = 2.5 R ⊙ ). Owens et al. [2008b] showed that F(r) increases with r, and the rise in F(r) À F S with r is what Lockwood et al. [2009bLockwood et al. [ , 2009c define as the "excess flux." Smith [2011] argued that excess flux is an unphysical artefact caused by the use of the modulus: a misconception that has been repeated in subsequent papers, even though in reality they are actually developing methods to deal with the same physical effects [e.g., Erdös and Balogh, 2014].
There are three key points that must be understood. First, excess flux, as defined by Lockwood et al. [2009bLockwood et al. [ , 2009c (from the modulus of the radial field component), is not unphysical. It arises from any mechanism that causes some heliospheric field lines to bend back toward the Sun such that for one crossing of the coronal source surface a field line can thread a heliocentric sphere of radius r three (and potentially even five or more) times. Such deviations from Parker spiral orientation can be associated with orthogardenhose IMF orientations on large spatial scales or with Alfvén waves on smaller scales. Excess flux is also caused by "folded flux" associated with interchange reconnection in dipolar streamers and pseudostreamers, and this folded flux can remain in gardenhose orientation .
modulus of the radial field) is independent of latitude and particle pressure gradients at greater r are too small to perturb this coronal imprint [Suess et al., 1996]. The constancy of |B r | is established by r ≈ 5 R ⊙ , i.e., very close to the coronal source surface and so the phenomena that cause excess flux act to perturb the radial constancy because they do not act equally at all latitudes. Lockwood and Owens [2009] have shown that if the excess flux is first removed and the latitudinal constancy then applied, the accuracy of the OSF value deduced using the Ulysses result is indeed increased.
The third key point is that although there are procedures that can be applied to modern satellite data to average out, reduce, or even eliminate excess flux, these are not possible for historical reconstructions from geomagnetic activity. One example of such a procedure is the sector averaging advocated by Smith [2011], which can be applied provided that electron heat flux measurements are available to define source sector boundaries with sufficient accuracy . Similarly, the methods of Erdös and Balogh [2014] can be applied if simultaneous high time-resolution local plasma velocity measurements are available. However, these options are not possible when we are dealing with reconstructions of near-Earth interplanetary conditions: in these cases, we can infer B which, using a solar wind speed estimate and Parker spiral theory, we can relate to |B r |-but as a consequence there is no option other than to quantify the excess flux and then subtract it. Hence, when comparing modeled OSF to reconstructions from geomagnetic activity, we must explicitly allow for the excess flux and use the formula (which is the definition of excess flux): where B rE is the near-Earth radial field and η is the IMF gardenhose angle, and ΔF is the excess flux caused by folds in heliospheric fields on all spatial scales. Because Parker spiral theory shows that η depends on the solar wind speed, equation (1) shows that there is not a simple relationship between the near-Earth IMF B and the OSF, F S : this relationship has been studied empirically by Lockwood et al.  Figure 1g are made using the approximate method of averaging B r over one day centred on the hour in question [Lockwood et al., 2009a]).

Streamer Belt Width
De Toma [2011] noted that in 2009 (during the long, low minimum between cycles 23 and 24), the near-Earth interplanetary satellites spent a larger fraction of the time in the slow solar wind than they had during previous solar minima in the space age. From eclipse observations of the solar corona, Cliver and Ling [2011] inferred that this was because the streamer belt was broader than during other recent solar minima and indeed showed that in images of the solar eclipse on 22 July 2009 the streamer belt appeared as wide as in images on 18 May 1901, a solar minimum in which geomagnetic reconstructions show the OSF fell to values very similar to those in 2009 [Lockwood et al., 1999[Lockwood et al., , 2009c[Lockwood et al., , 2014d]. [2014] recently used the reconstructions of solar wind speed shown in Figure 1d to discuss the origins of the much lower average slow solar wind speeds derived for the solar minima around 1879 and 1901 (between cycles 11 and 12, and between cycles 13 and 14, respectively). They estimated the latitudinal width of the streamer belt and inferred that it was exceptionally wide during these minima and concluded that the low solar average solar wind speed occurred because the Earth remained consistently within the streamer belt during these minima. Figure 1d shows that the annual means of V SW in these two minima are very close to the lowest values of the distribution of hourly means observed by near-Earth satellites shown by the grey histogram in Figure 1e. From this similarity, Lockwood and Owens [2014] suggest that the speed of the slow solar wind is unchanged but the average solar wind speed < V SW > at Earth is low because Earth resides almost continuously in the band of slow solar wind, rather than because of a change in the fundamental properties of the fast or slow solar wind. Evidence to support this idea comes from the recent solar minimum between cycles 23 and 24. That the streamer belt width was greater during minimum 23/24 has also been inferred from the Thompson-scattered light from the solar corona, as detected by coronagraphs, and from interplanetary scintillation observations of the solar wind [Manoharan, 2012]. Lockwood and Owens [2014] used the method of Owens et al. [2013Owens et al. [ , 2014 to define the streamer belt width by identifying dipole streamers and pseudostreamers using magnetograph data and PFSS modeling.

Lockwood and Owens
Both types of streamer act as sources of the slow solar wind [Antiochos et al., 2011;Riley and Luhmann, 2012;Owens et al., 2013Owens et al., , 2014, and Lockwood and Owens [2014] showed that this belt decreased in latitudinal width over all three declining phases of the observed solar cycles but did not become as thin during minimum 23/24 as it had at the previous two minima.  Lockwood et al., 2014b] discussed the divergence of the (corrected) international sunspot number and the group sunspot number in the years before the start of the photographic observations by the Royal Greenwich Observatory in 1874. Because there is no way to know which of these two sunspot records best quantifies OSF emergence in these years, in the present paper, we use both the group sunspot number R G and the (corrected and extended) international/Zürich/Wolf sunspot number composite R C and discuss the implications for the modeling of the differences between the two.

Relationship Between Reconstructed Solar Wind Velocity and Open Solar Flux
In a series of four papers, Lockwood et al. [2013aLockwood et al. [ , 2013bLockwood et al. [ , 2014aLockwood et al. [ , 2014b have recently extended the aa index and the OSF, near-Earth IMF, and solar wind speed reconstructions back to 1845, as shown by Figure 1. The longest reconstructions are based on two pairings, the corrected aa index, (aa C ) with the IDV index and aa C with the IDV(1d) index. The results are very similar after 1872 and the tests presented in Papers 1 and 2 show that where they do differ before this date, it is the IDV index that is behaving differently. Thus, for 1845-1872 the reconstructions for aa C and IDV(1d) are the most reliable. effect has been allowed for in the F S values, and Figure 2 shows that the relationship between V SW and F S cannot be explained by an overestimation of this effect. Hence, average solar wind speeds at Earth during solar minimum are genuinely lower when the OSF is low. In the remainder of this paper we model the streamer belt width to analyze the concept of Lockwood and Owens [2014] that this is associated with increased width of the streamer belt at sunspot minimum, and to extend this understanding to the Maunder minimum.

The Basic Equations
Following Solanki et al. [2000], we model the signed OSF, F S (here defined as the flux of one polarity threading the coronal source surface) using the global solar continuity equation where S is the source rate and f is the fractional OSF loss rate (so that the total loss rate is f × F S ) which, following , we generalize to be a function of solar cycle phase, Φ, only. This quantifies the degree of heliospheric current sheet (HCS) warping and its effect on the OSF disconnection rate.

Owens and Crooker
[2007] generated a more complex formulation of the OSF budget using OSF subdivisions based on different suprathermal electron populations, and this was employed by Schwadron et al. [2010] who added an explicit time constant for converting "closed" heliospheric loops to "open" coronal hole flux (rather than using the assumption of Owens and Crooker that the counterstreaming signature is lost within about 1 month of emergence) and allowed for separate disconnection and interchange reconnections. This provides a very elegant formulation of the OSF budget, but when it comes to fitting the observed and reconstructed OSF suffers from the larger number of free parameters, as discussed in section 1.3, and which are not essential for the working definition of OSF, as discussed in section 1.2. Instead of building up to a total OSF solution using the subdivision and equations of Schwadron et al. [2010], we here adopt the simpler formulations of production and loss rates of  with its single free fit parameter to compute the total OSF. However, we then investigate the suggestion of Lockwood and Owens [2014] by applying some of the principles discussed by Schwadron et al. [2010] to subdivide the OSF into two categories: streamer belt (SB) flux and coronal hole (CH) flux, the latter including both polar coronal holes and coronal holes at lower latitudes. Hence, We assume that all newly emerged OSF is injected into the SB (i.e., the total flux emergence rate through the coronal source surface S is source term for SB flux). Some of this emerged flux is lost by disconnection at reconnection X-lines in the HCS which is embedded in the SB. The remainder that stays connected evolves into CH flux as the flux tube expands into the heliosphere and its foot points migrate poleward across the Sun under the influence of meridional flow and interchange reconnections. Given that flux tubes reach the termination shock within about a year of emergence, after which no information about the flux tube evolution can be received in the inner heliosphere, this evolution from SB to CH on timescales greater than a year is associated with flux transport in the photosphere and not evolution of the OSF flux loop in the heliosphere. Interchange reconnections move the foot points of the CH flux, but its loss requires reconnection between adjacent CH field lines of opposite polarity. From these considerations, the continuity equation for SB flux is: where f SB × F S is the rate of disconnection of SB flux at the HCS and S CH is the rate at which SB flux is transferred to coronal hole flux. The corresponding continuity equation for CH flux is Note that f SB and f CH are the "fractional" loss rates, defined as the disconnection rate (for SB and CH flux, respectively) per unit total OSF. Adding equations (4) and (5), using equation (3) and comparing with equation (2) gives Hence, the total fractional loss rate can be divided into part taking place in the streamer belt and part taking place in coronal holes [Schrijver et al., 2002].  used the reconstructed open solar flux variation with equation (2) to show that the fractional loss rate depended only on solar cycle phase and that the cyclic part was highly correlated with HCS tilt, as expected theoretically for the disconnection of SB flux [Owens et al., 2011b].
In order to solve these equations we have to know the variation in the rate at which SB flux evolves into CH flux, giving the loss term S CH in equation (4). Because all newly emerged OSF is envisaged as being injected into the SB, S CH is the only source term in equation (5). The rate S CH can be computed for a given temporal variation in S with a few simple assumptions, as discussed in section 3.3 below.
Once the total open solar flux F S and streamer belt flux F SB have been computed, the latitudinal half width of the streamer belt Λ SB can be computed because of the latitudinal constancy of the radial component of the heliospheric field giving the equation used by Lockwood and Owens [2014]: 3.

Input Emergence Rate Variation
The only source of information on the long-term variation in OSF emergence available to us is the sunspot number record, and Solanki et al. [2000] demonstrated that it could be used in the OSF continuity equation. We here use both the international/Zürich/Wolf sunspot number composite and group sunspot number sequences, the former having been corrected to allow for calibration problems around the "Waldmeier" and "Wolf" discontinuities, as discussed in Papers 1 and 2. As noted in Paper 2, the corrected international sunspot number, R C , and the group sunspot number, R G , diverge as we go back in time-a divergence that is present after 1874 (the start of the RGO data) but is most pronounced before then but which fluctuates with time. There is no way to say if R C or R G better describes the OSF emergence rate before the mid-19 th century, and so we here consider the implications of both. The R G sequence extends back to 1610, before the start of the Maunder minimum, and we here extend R C back to the same starting date using 1.3R G , as proposed in Paper 2.
From estimates of the CME occurrence rate over recent solar cycles and the magnetic flux emerging into the SB,  derived the flux emergence rate per year. We employ a slight variant of that equation (based on the best fit between CME rates and sunspot numbers [Owens et al., 2008c]) to predict the total OSF emergence rate: where F CME is the estimated average signed flux content injected by a CME, and R is the sunspot number estimate (we employ R C and R G ). We here use F CME = 3.5 × 10 11 Wb [Owens et al., 2008c]. The terms a and b give a correction to allow for the fact that we do not know how much smaller the emergence rate in the Maunder minimum S MM was, compared to the baselevel detected in the recent low solar minimum, S o . The term in square brackets is the rate used by  (with slightly changed coefficients obtained by fitting to a slightly updated data set) which assumes that the recent minimum set a baselevel emergence rate that also existed during the Maunder minimum (i.e., they assumed S MM = S o ). Hence, the use of b = 0 and a = 1 recovers this assumption. The largest possible value for b = S o À S MM is 1.43 × 10 14 Wb yr À1 (because any larger b would result in negative S MM ). To give an uncertainty analysis, we varied b over the full range up to this maximum value. The best fit of the modeled F S variation to that derived from igeomagnetic observations was found to be for b = 0 (and a = 0.95). The uncertainty in this value and its effects will be discussed later. The best fit variation in the OSF source rate, S, from equation (6) using these best fit a and b values with R C and R G is shown by the black and mauve lines, respectively, in Figure 3.

The Transfer of Streamer Belt Flux to Coronal Holes
In order to compute the rate of flux transfer S CH from the streamer belt to the coronal holes, we assume that the loss of SB flux by disconnection occurs during the first year after emergence (after that foot points are deemed too have migrated poleward too far from HCS for such disconnection to happen). We also allow for the SB flux evolving into CH flux on a distribution of timescales. We here assume that of the emerged flux SB that survives the firstyear disconnection, 10%, 50%, 30%, and 10% are transferred to CH flux after y = 1, 2, 3, and 4 years, respectively, following emergence. Hence, after 4 years all emerged SB flux has either been disconnected or evolved into CH, and the average time for this evolution is 2.4 years. This evolution profile was derived iteratively, but the results were largely insensitive to the precise profile as long as the mean value remained close to 2.4 years (the main effect of the distribution to be to add some smoothing to the modeled F CH and F SB sequences). Thus, S CH can be computed for a given variation of the emergence rate S, provided that the loss of that emerged flux in the first year subsequent to emergence (f SB × F S ) is known: From the above, we employ the temporal transfer profile of w(t À 1) = 0.1, w(t À 2) = 0.5, w(t À 3) = 0.3, w (t À 4) = 0.1, and w(t À 5) = 0, giving the mean time before transfer of 2.4 years. The average was adjusted to 2.4 years to make the predicted F CH in phase with the solar polar fields measured by the Wilcox Solar Observatory (WSO) magnetograph (see discussion below about Figure 8). We note that Schwadron et al.

OSF Loss Rate Variation
Using the reconstructed OSF variation and a source rate variation similar to that shown in Figure 3,  noted that the fractional loss rate f required to model the data using equation (1)  The bottom panel of Figure 5 shows the fractional loss rate f required to match the OSF variation for the source rate variation S prescribed by Figure 4. The blue and red dots are derived using equation (2)  and  from the geomagnetic OSF reconstruction data: blue dots are the results if R C is used to compute S and red dots if R G is used. The OSF data have been passed through a three-point running mean to smooth out fluctuations. The black line is the fit to the average variation with solar cycle phase Φ, plotted as a function of time using the variation of Φ derived from < |Λ H | >.
Also shown in the bottom panel of Figure 4 (in green) is the best fit linear regression of the Heliospheric Current Sheet index [Owens et al., 2011b] derived from application of the PFSS method to the WSO magnetograph data. It can be seen that the variation of this index with Φ is very similar to that of the average fractional loss rate. This gives strong support to the idea that the cycling of the fractional loss rate, shown by the blue and red dots in Figure 4c, is caused by the disconnection of SB flux by reconnection in the HCS, as proposed by Owens et al. [2011b]. Thus, this cyclic variation is best explained as a feature of the loss of SB flux. However, there is a baselevel fractional loss rate f of about 0.8 year À1 , some of which could be baselevel SB disconnection, and some of which is disconnection of CH flux [Schrijver et al., 2002].
Given that the cyclic behavior of the total fractional loss rate f is well explained by its streamer belt component f SB (see equation (6)) there is no evidence that the coronal hole component f CH is not constant. However, the value of f CH is not known. We must have f CH > 0 or, by equation (4) coronal hole flux will continuously grow and the streamer belt disappear. From Figure 5 we know f CH < 0.8 year À1 , the minimum value of f seen every solar cycle. To determine the optimum f CH , we vary it between these two limits and use the equations of section 3.1 to generate the time series for 1610-2012 of the OSF F S , the SB flux F SB , the CH flux F CH , the latitudinal half width of the streamer belt Λ SB , and the transfer rate between the streamer belt and coronal holes, S CH . Note that the solar cycle phase Φ was determined for solar cycles 8-11 using the extension of the butterfly diagram by Arlt et al. [2013]. Before 1825 there is no continuous record of sunspot latitudes from which to determine the variation of Φ, and so the times of the minimum in R G are used to define the times of Φ = 0 and a linear variation of Φ with time assumed in each cycle. During the Maunder minimum the maxima in the cycles of 10Be isotope abundance are used [Beer et al., 1998], as in the study by .
The parameter f CH is the major free parameter in modeling the streamer belt width presented in this paper, and Figure 5 explains how the value of this parameter is set. Figure 5 shows the results for f CH values in the allowed range between zero and 0.8 year À1 . Using both R C and R G , the initial results show some influence of the initial (1610) values of F S and F SB (and hence F CH ) from which the solution of equations (1), (3), and (4) are initiated. However, by 1620 all influence of these spin-up values was found to be lost. The black line shows the fraction of the 391 annual values after 1620 for which F SB (and hence Λ SB ) fall to zero, P[Λ SB = 0], i.e., when the streamer belt disappears. As expected, if f CH is too small, this happens quite frequently. It is also found to occur increasingly often if f CH is too large because this causes f SB to be too small and too much flux to be transferred to the coronal holes, i.e., S CH is too large. Only for the limited range 0.18 < f CH < 0.37 year À1 does the streamer belt not regularly vanish. The best determination of f CH comes from comparing the latitudinal half width of the streamer belt Λ SB in recent minima with that obtained from the occurrence frequency of dipole streamers (DS) and pseudostreamers (PS) determined from PFSS maps of the solar corona from magnetograph data using the method of Owens et al. [2013Owens et al. [ , 2014. This is discussed further later in the present paper. The dashed horizontal lines in Figure 5 are the minima of the streamer belt half-width Λ SB for the solar minima between cycles 22 and 23 (mauve dashed line) and between cycles 23 and 24 (blue dashed line), as determined from where DS and PS streamer frequency falls to zero. The solid lines of the same colors show the model predictions for Λ SB at these minima as a function of f CH . In both cases, the values closest to the observed values occur for f CH = 0.22 year À1 . Thus, we adopt f CH = 0.22 year À1 as the value giving the best match to the SB width during recent solar minima.
The green line in Figure 5 shows another consequence of the f CH value adopted by showing the fraction of the 391 annual values of S CH that are negative, P[S CH < 0] which we would not expect to occur. The number of years in which this happens declines with f CH and is zero for f CH > 0.32. In fact in this regime, S CH is generally too large, leading to non-zero P[Λ SB < 0] which increases with f CH . The red and cyan lines in Figure 3 show the variation of S CH for the optimum f CH of 0.22 year À1 : it can be seen that negative values are seen before 1620 which are associated with initial starting conditions and the model spin-up. Subsequently, negative S CH occurs just 3 times and then by small amounts which we can attribute to the numerical uncertainties in the model. Hence, although negative S CH is a problem if f CH is too small, it is at an acceptable level at f CH = 0.22 year À1 .

Model Results
Setting the fractional coronal hole loss rate f CH allows equations (2)-(7) and (9) to be solved for a given emergence rate variation S(t). In particular, the variations in the total OSF, F S , the coronal hole OSF, F CH , the streamer belt OSF, F SB , and the latitudinal half width of the streamer belt, Λ SB , can all be calculated. We use S(t) given by equation (8) using both the corrected and extended international sunspot number, R C and the group number, R G . By using a range of these starting values we find that all influence on any of the modeled parameters is lost within one solar cycle, i.e., by 1620, and hence we regard the first 10 years as model spin-up time only. Figure 6 shows the variation of modeled total OSF, F S , for R C in blue and R G in red. Note that these simulations do not depend on the division of OSF between the SB and CH flux, being determined by equation (2) which uses only the total fractional loss rate, f, which is set by solar cycle phase only, as demonstrated by Figure 4. Hence, these results are not dependent on the derived value for f CH discussed above. The black line in Figure 6 is the reconstructed OSF variation from geomagnetic data (that derived using IDV(1d) and aa C is shown, but as demonstrated by Figure 1, other combinations of indices give almost identical results). Remember that the input variation of the emergence rate S has been fine-tuned via the parameter b to give the best fit (lowest RMS deviation) which yields a correlation coefficient of 0.873. This fit was for b = 0. The variation of the RMS fit residual for full range of potential b values was evaluated. The probability density function (pdf) of the differences between the minimum fit residual and those at general b were evaluated and those within the 2σ significance level gave b in the range À0.05 × 10 14 Wb yr À1 ≤ b ≤ +0.11 Wb yr À1 . This sets the 2σ uncertainty ranges of modeled F S values in any year shown by the gray band. The blue and red lines show the results using R C and R G , respectively, and the uncertainty band for R G (not shown for clarity) is very similar in width to that for R C (shown in gray) in all years. The black line lies within this gray band for all but 12 of the 167 data points, the most consistent failure to fit being at the peak of solar cycle 20 (around 1970) when modeled values are consistently higher using both R C and R G .

Total Open Solar Flux
The derived OSF during the Maunder minimum is interesting. Averaging over a whole number of the cycles in the MM (1655-1696) gives a mean OSF of (0.120 ± 0.061) × 10 À15 Wb. If we optimize b by maximizing the correlation coefficient between modeled and reconstructed OSF (rather than minimizing the RMS residual) we get a somewhat larger optimum b of 0.15 Wb yr À1 which reduces the MM OSF prediction somewhat to (0.057 ± 0.041) × 10 À15 Wb: a lower value but the uncertainty ranges of the two estimates do overlap. The value obtained by Lockwood and Owens [2011] from the estimate of the near-Earth IMF from cosmogenic isotopes by Steinhilber et al. [2010] was (0.048 ± 0.029) × 10 15 Wb. The modeling of OSF by Lockwood [2003] (running backward in time from an initial value taken from the first Ulysses perihelion pass) gave an estimate of 0.1 × 10 15 Wb. Cliver and Ling [2011] argue that there is a floor value of the OSF of 0.080 × 10 15 Wb. The minimum values of the cycles modeled here for during the Maunder minimum are 0.075 × 10 15 Wb with the 2σ uncertainty limits at 0.04 × 10 15 Wb and 0.13 × 10 15 Wb. Thus, the modeling here suggests that the OSF during the Maunder minimum fell close to Cliver and Ling's proposed floor value. We would argue that this is still not a floor in any meaningful sense. From equation (1) the minimum OSF is simply the ratio of the minimum emergence rate S divided by the maximum fraction loss rate f. We here find no physical reason to set a firm minimum to S, so there is no (Note that for many years the red line is not visible as it is overlaid by the blue line). The black line is the reconstruction of F S made using the aa C and IDV(1d) geomagnetic indices (see Figure 1). All three series have been smoothed with a three-point running mean to improve plot clarity. The correlation between the modeled and reconstructed data is 0.876. The gray area is bounded by the maximum and minimum values for each year of all fits for which the RMS deviation is different from the optimum at the 2σ level. physical limit (a "floor"): it is simply that the probability of a given OSF is decreasingly small as OSF decreases in this tail of this distribution.
It should also be noted that the agreement between the lowest OSF estimates presented here and the proposed floor value in Cliver and Ling [2011] is possibly somewhat coincidental. Cliver and Ling argue that the near-Earth IMF comprises a 2.8 nT floor caused by magnetic field embedded in the slow solar wind (the SB flux) and a variable part that varies between 0 and 3 nT from cycle to cycle associated with the solar polar fields (the CH flux). Figure 7 shows a radically different prediction from the modeling presented here. The SB flux is far from constant; indeed, its variations are the dominant cause of the OSF variation as at most times the OSF is dominated by SB flux (the CH flux is only larger in the late declining phase and minimum of each cycle). As a result, the OSF peaks around sunspot maximum (and not near sunspot minimum when CH flux peaks). It is important to understand that B and |B r | at Earth depend on both the CH and SB flux, irrespective of whether Earth is within the streamer belt of not. This is because the latitudinal constancy of |B r | is established very close to the Sun (within r ≈ 5 R ⊙ [Suess et al., 1996]) so that, as pointed out by Lockwood and Owens [2014], although the width of the streamer belt depends on the ratio of the SB and CH fluxes, |B r | (and thence B) at any heliographic latitude (inside the streamer belt or not) depends on the sum of the SB and CH fluxes.
The effect of using the group sunspot number R G instead of R C can be seen in Figure 6. Because they are so similar for most of the interval of the geomagnetic OSF reconstruction, the results over this interval are very similar indeed. As expected, predicted OSF values are lower on either side of the Dalton minimum if R G is used rather than R C . Within the Dalton minimum, cycle amplitudes are larger if R C is used, but the cycle minima are very similar for R C and R G . Also, as expected from Paper 2, the fits to the reconstructed OSF are better in cycle 9 using R G but better in cycle 11 using R C .

Streamer Belt and Coronal Hole Flux
The top panel of Figure 7 shows the variations of the coronal hole (in blue) and streamer belt (in red) fluxes derived using R C and the optimum f CH of 0.22 year À1 . The area shaded gray gives the sum of the two, which is the total OSF, F S . It can be seen that the peak F S values (around sunspot maximum) are dominated by F SB , and F CH at these times is a minimum for each cycle. At the times of the minima in F S , F SB is also at a minimum and F CH at a maximum giving an almost antiphase behavior of F SB and F CH . This behavior is reproduced for any profile of the rate of evolution of SB flux into CH flux (the w(t-y) coefficients in equation (9)) that ensures the peak transfer occurs between 2 and 4 years after emergence. With this phase lag over each solar cycle, the long-term variation in F CH follows that in F S quite closely. The plot corresponding to Figure 7a generated using R G is similar in every way, other than all fluxes are lower on either side of the Dalton minimum, as seen for F S in Figure 6.

Streamer Belt Width
Using equation (7), the variation of the width of the streamer belt can be computed from F S and F CH . The lower panel of figure 7 shows the modeled variation of the sine of the latitudinal half-width of the streamer belt, sin(Λ SB ). The black line uses R C to quantify the OSF emergence rate whereas the mauve line uses R G . It can be seen that the model predicts times when the coronal hole flux goes to zero (sin(Λ SB ) = 1), but there are no years when the streamer belt vanished (sin(Λ SB ) = 0). Very thin streamer belts (sin(Λ SB ) < 0.3) in Figure 8 always precede a year of with low transfer rate ( Figure 5) and arise in sunspot minimum years for which the 11 year running mean of sunspot number is high (<R C > 22 > 55). Results for R C and R G are very similar at each solar maximum when sin(Λ SB ) peaks. However, the minima in sin(Λ SB ) are less deep using R G (compared to those for R C ) on either side of the Dalton minimum when R G also yields lower F S values.
The best test of the streamer belt width modeling is to compare with the streamer belt width deduced over recent solar cycles using Potential Field Source Surface (PFSS) modeling of coronal fields from magnetograph observations of photospheric magnetic fields. From the photospheric separation of adjacent field lines at the coronal source surface (at an heliocentric distance of 2.5 R ⊙ , where R ⊙ is a mean solar radius), Owens et al. [2013] define the occurrence of dipolar streamers (DS) and pseudostreamers (PS). The colored pixels in Figure 11 show the sum of the occurrence frequencies of DS and PS (P PS + P DS ) as a function of the sine of the heliographic latitude, sin(Λ H ), and time (bins are 0.044 by 1 year in size). Black areas, where (P PS + P DS ) = 0 are polar coronal holes. The occurrence of DP or PS is taken to be an indicator of the SB. The blue lines show the variation of the SB edges predicted by the model (assuming that it is symmetrical so the half-width Λ SB applies in both solar hemispheres). It can be seen that the model has captured both the solar cycle variations and the cycle-to-cycle changes in the minimum SB width well.
The upper panel of Figure 7 is a check on the phasing of the derived F CH variation and the use of an average transfer time of surviving SB flux to coronal holes of 2.4 years. The blue line is the modeled F CH for the last three solar cycles, and the red line gives the variation of the solar polar field observed by the WSO magnetograph (scaled so it can be shown on the same axes). This field is for < B p > = (B N + B S )/2 where B N and B S are the fields detected over the north and south solar poles, respectively. We would not expect an exact correspondence as F CH is flux and not a field magnitude, and so at solar minimum the variations in the size of the coronal holes are a factor and at sunspot maximum coronal holes occur all latitudes. Nevertheless, the model has produced a variation in F CH that is in phase with the observed < B p > variation, and the decline in the peak values of < B p > over the last three solar minima is reflected in a decline in the corresponding modeled F CH peaks. Figure 7b shows that the model is predicting a persistently broader SB during the Maunder minimum, as suggested by Lockwood and Owens [2014]. Figure 9 plots the modeled latitudinal half-width of the streamer belt (SB), Λ SB , as a function of the open solar flux (OSF) for 1620-2012. Black points in this plot are based on the sunspot numbers series R C , and mauve points are based on the group sunspot number R G . The plot is in the same format as Figure 2 with annual modeled points for the relevant sunspot number less than 20 shown by filled circles and for sunspots numbers greater than or equal to 20 shown by open circles. Figure 9 has the general approximate appearance of an inverted version of Figure 2, with the notable feature that the modeled streamer belt width at sunspot minimum increases as OSF decreases for the low sunspot number data. This gives strong support to the idea proposed by Lockwood and Owens [2014] that annual means of sunspot minimum solar wind velocities fall to the values of the slow solar wind when OSF is small because the streamer belt is wide and Earth remains immersed in the streamer belt and slow (or perhaps intermediate speed) solar wind flows.

Discussion and Conclusions
The fraction of the OSF lost from coronal holes (CH) flux is best quantified by matching the predicted streamer belt width to the variations deduced for recent solar cycles using the PFSS method. The value derived sits comfortably within the allowed range between two model behaviors that are deemed unphysical: namely, persistent loss of all SB flux and net migration of CH flux back into the SB. The OSF loss rate is always dominated by the SB flux loss.
The modeled OSF is dominated at most times by the SB flux. The modeled CH flux varies in close to antiphase with the SB flux (it lags by a bit less than half a cycle) over the solar cycle and shows a cycle-to-cycle variation that mirrors in form that in the SB flux and total OSF. The modeled CH flux variation captures the phase and recent declining amplitude trend of the observed variation in solar polar fields. In the Maunder minimum cycles of OSF are deduced despite the fact that the input open flux emergence rate is at a steady baselevel value. These cycles have been explained by  as the effect of the cyclic SB loss, whereas for larger sunspot number cycles the emergence rate cycles dominate the OSF behavior. During the Maunder minimum, the modeled CH flux falls to zero at each maximum of the OSF cycles, but the CH and SB fluxes are close to equal at each of the MM minima in OSF.
The use of corrected international sunspot number R C instead of group sunspot number R G makes no difference in principle to the behavior but does make some difference to the detail of the variations. As would be expected, the OSF is larger on either side of the Dalton minimum if R C instead of R G is employed. The difference within the Dalton minimum itself is much smaller. The width of the SB reaches the same maxima for R C and R G at these times, but the minima seen at sunspot minimum are less deep for R G than for R C . This is consistent with the general trend for the sunspot-minimum SB width to increase for lower OSF values. The model outputs (the streamer-belt half-width and the coronal-hole, streamer-belt, and total open solar fluxes) are available in the supplementary data file attached to the present paper. The input sunspot number data are available in the supplementary data file attached to paper 2 [Lockwood et al., 2014b]. The open solar flux, solar wind speed and near-Earth IMF reconstructions from historic geomagnetic data are available in the supplementary data file attached to the paper by Lockwood et al. [2014d].