Excess open solar magnetic flux from satellite data: 2. A survey of kinematic effects


  • M. Lockwood,

    1. Space Environment Physics, School of Physics and Astronomy, Southampton University, Southampton, UK
    2. Also at Space Science and Technology Department, Rutherford Appleton Laboratory, Chilton, UK.
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  • M. Owens,

    1. Space and Atmospheric Physics Group, Blackett Laboratory, Imperial College London, London, UK
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  • A. P. Rouillard

    1. Space Environment Physics, School of Physics and Astronomy, Southampton University, Southampton, UK
    2. Also at Space Science and Technology Department, Rutherford Appleton Laboratory, Chilton, UK.
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[1] We investigate the “flux excess” effect, whereby open solar flux estimates from spacecraft increase with increasing heliocentric distance. We analyze the kinematic effect on these open solar flux estimates of large-scale longitudinal structure in the solar wind flow, with particular emphasis on correcting estimates made using data from near-Earth satellites. We show that scatter, but no net bias, is introduced by the kinematic “bunching effect” on sampling and that this is true for both compression and rarefaction regions. The observed flux excesses, as a function of heliocentric distance, are shown to be consistent with open solar flux estimates from solar magnetograms made using the potential field source surface method and are well explained by the kinematic effect of solar wind speed variations on the frozen-in heliospheric field. Applying this kinematic correction to the Omni-2 interplanetary data set shows that the open solar flux at solar minimum fell from an annual mean of 3.82 × 1016 Wb in 1987 to close to half that value (1.98 × 1016 Wb) in 2007, making the fall in the minimum value over the last two solar cycles considerably faster than the rise inferred from geomagnetic activity observations over four solar cycles in the first half of the 20th century.

1. Introduction

[2] In our companion paper [Lockwood et al., 2009] (hereinafter referred to as paper 1), we discussed the Ulysses result of the latitudinal invariance of the radial heliospheric field and its relationship to the “flux excess” detected by Owens et al. [2008a]. As given in paper 1, using the latitudinal invariance gives the signed open flux (of one polarity), FS to be

equation image

where r is the heliospheric distance and Br is the radial component of the heliospheric field. The subscript CR is to denote that the averages are taken over a Carrington Rotation interval (or alternatively a Bartels rotation interval) to remove longitudinal structure. T is the timescale on which Br data are preaveraged and then converted into absolute values. Lockwood et al. [2004] showed that the error introduced into FS by use of equation (1) was <5% if averages over a 27 days or longer are taken: this was shown to be true for both solar maximum and solar minimum conditions.

[3] Recently Owens et al. [2008a] have studied the open solar flux (in fact, they studied the unsigned flux; i.e., 2FS) deduced from spacecraft in different parts of the heliosphere using T = 1 h. They find considerable agreement between the data sequences from different craft which gives strong support to the use of equation (1). However, although they find that neither latitudinal nor longitudinal separation of the craft introduced significant differences on average, they did find a consistent increase in the estimated FS with heliocentric distance r, which became especially pronounced at above about 2.5 AU. This “flux excess” effect was discussed in relation to the third Ulysses perihelion pass in paper 1. Although the survey by Owens et al. [2008a] did not find a consistent variation of the excess flux with heliographic latitude Λ, these authors noted that any variation with Λ in Ulysses data was convolved with the dependence on r owing to the nature of the satellite's orbit. In fact, these data are further complicated by the sunspot cycle phase ɛ which changes on timescales comparable to the Ulysses orbital period; separation of the effects of r, Λ and ɛ are presented by Lockwood and Owens [2009]. In paper 1 we report a difference between the excess flux in the streamer belt and the large polar coronal holes for the third pass of Ulysses which was near solar minimum.

[4] The open solar flux (here defined as the flux threading the coronal source surface) has also been evaluated from solar magnetograms using the potential field source surface (PFSS) method [e.g., Schatten et al., 1969; Altschuler and Newkirk, 1969; Schatten, 1999]. In this method, the observed photospheric magnetic fields from a magnetogram are mapped through the solar corona to the source surface (assumed to be spherical with radius r = ro = 2.5 solar radii = (2.5/251)AU, where 1 AU ≈ 1.5 × 1011 m) with a number of assumptions. Agreement with data from in situ observations of the heliospheric field has generally been good [Wang and Sheeley, 2003], but two caveats to this comparison should be noted. First, the magnetogram data require processing using a latitude-dependent instrument saturation factor [Wang and Sheeley, 1995] which has been the subject of some debate [Svalgaard et al., 1978; Ulrich, 1992]. Second, for the satellite data taken near Earth (at heliocentric distance r ≈ 1 AU), preaveraging of Br on timescales of T = 1–2 days (before absolute values are taken) has been required: otherwise the latter would show a flux excess (relative to the PFSS values). As discussed in paper 1, the idea is that the value of T is chosen so that it is not so large that the opposing field in “toward” and “away” interplanetary sectors of the source field are canceled (which would cause the true open solar flux to be underestimated) but should be large enough that small-scale structure generated in the heliosphere (which does not reflect structure in the source field and so would cause the true open flux to be overestimated) is averaged out. Increasing T results in lower FS values given by equation (1) [Lockwood et al., 2006], but there is no a priori reason for adopting any one value of T. At any one time, there will always be a T for which the near-Earth estimate using equation (1) will equal the true coronal source flux, but without an understanding of how to compute it, we cannot be sure of the correct T to use nor if it is constant with either time or heliographic coordinates. Indeed, it may be that the heliospheric effects cannot be separated from source sector structure by their timescales, in which case the use of T has no physical justification at all. A value for T of 1 day has commonly been adopted because it has given a good general match of the variations in FS deduced from near-Earth in situ data to those from the PFSS method [Wang and Sheeley, 1995; Wang et al., 2000; Lockwood et al., 2006].

[5] Paper 1 analyzed the effect of various T, and of plasma time-of-flight associated with large-scale longitudinal solar wind variability (“kinematic effects”; see section 2), on the flux excess detected by comparison of data from the near-Earth ACE satellite with that from the Ulysses spacecraft during its third perihelion pass. Paper 1 found that both effects provided a partial allowance for the flux excess, although the kinematic correction performed more satisfactorily in a number of respects. However, it also showed that these are not equivalent corrections in that the kinematic correction is allowing for larger-scale structure in the heliosphere (giving variations at any one point on timescales 1–27 days) whereas the averaging timescale can only make allowance for smaller-scale structure (on timescales of T ≤ 1 day). Furthermore, paper 1 showed that structure on timescales T < 1 h was not contributing to the observed flux excess because its effect was the same at all r and Λ. The averaging and the kinematic corrections generated somewhat different estimates of the open solar flux FS. The kinematic correction gave a more uniform variation of the radial field with latitude and gave closer agreement between the ACE and Ulysses data (but not by an overwhelmingly large factor). One major advantage of the kinematic correction was that it matched a difference between the streamer belt and the sunspot-minimum polar coronal hole, whereas averaging over a fixed time interval T did not. In the present paper, we consider the kinematic correction of coronal source flux estimates from r near 1 AU and elsewhere in the heliosphere and show it to be consistent, not only with the flux excess variation as a function of r in the inner heliosphere as deduced by Owens et al. [2008a], but also with PFSS open flux estimates.

2. Theory of Kinematic Effects

[6] Figure 1 is a schematic of how large-scale temporal (Figures 1a1c) and spatial (Figures 1d1f) structure in the radial solar wind flow speed can give rise to excess flux via purely kinematic (time-of-flight) effects and how this excess flux grows with heliocentric distance r. In both cases, this arises because the field becomes distorted as it is frozen into the longitudinally variable flow. Both spatial and temporal effects result in the excess flux in measurements made at rr1 = 1 AU which therefore need correcting for these kinematic effects. As suggested by comparison of Figures 1c and 1f, it will not be easy to distinguish the spatial and temporal effects in many cases, as both result in an amplification of the magnitude of the ambient radial field in regions of increasing solar wind speed V (dV > 0) and a reduction (and, at sufficiently large r, a reversal in polarity) in regions of decreasing solar wind speed (dV < 0). Examples of such kinematic effects have been reported in the streamer belt: Riley and Gosling [2007] have shown that events of near-radial IMF reported by Jones et al. [1998] are explained by the kinematic effect in rarefaction regions where the solar wind velocity decays (dV < 0). Burlaga and Barouch [1976] provided equations which quantify the effects shown schematically in Figure 1. It is important to note that these equations do not allow for dynamical stream–stream interactions that will inevitably accompany the kinematic effects associated with increasing solar wind flow speeds in compression regions (dV > 0) [Gosling, 1996; Gosling and Pizzo, 1999; Arge and Pizzo, 2000]. Compression regions steepen as they propagate to greater r, until shocks form. However, the lack of dynamical effects in rarefaction regions means that the kinematic effects will grow with increasing r until the IMF becomes radial [Jones et al., 1998; Riley and Gosling, 2007].

Figure 1.

Schematic of the evolution of an interplanetary field line for (a–c) a transient flow speed increase and (d–f) a long-lived rotating fast flow region threaded by a seed tangential field. In both cases the fast flow region is shaded and the growing distortion of the field line (with respect to a Parker spiral shown by the dotted line) makes an additional contribution to the mean radial field (which averaged over a full Bartels rotation is 〈∣ΔBr∣〉27), and this additional contribution grows with increasing heliocentric distance r.

[7] Burlaga and Barouch [1976, equation (16)] gives the vector field at a heliocentric distance r for purely kinematic effects:

equation image


equation image

Bro and equation image are the radial and longitudinal components of the field at the coronal source surface r = ro, λ is the heliographic latitude, Ω is the angular rotation velocity of the solar atmosphere, V is the (radial) solar wind speed and μ is a dimensionless parameter that allows for the kinematic steepening of gradients owing to time-of-flight effects: it therefore depends on r. [∂V/∂t]o is the temporal solar wind velocity gradient at r = ro. Equation (3) is also readily derived by applying the frozen-in theorem to two plasma elements on the same field line but on solar wind streamlines of different speed. As pointed out by Burlaga and Barouch, for uniform solar wind speed [∂V/∂t]o = 0, this equation reduces to the standard Parker spiral equation. Note that the “seed” tangential field Bφo is a constant for a given field line. From equation (2) the magnitude of the equation image component at general r is

equation image

Substituting (4) and (3) gives

equation image

If we consider initially only kinematic effects; that is, with no modification of the solar wind velocity vector V by stream–stream interactions, the radial velocities V and V + dV observed at r (at times t and dt) will be the same for those plasma parcels at all r. Plasma parcels seen dt apart at r will have left the coronal source surface at an interval dto apart. Defining the times-of-flight for the radial velocities V and V + dV to be τV and τV+dV:

equation image

Thus [∂V/∂t]o = dV/dto can be computed as all the terms on the right-hand side of equation (5) are known from the spacecraft data at r. Hence ΔBr can be predicted from solar wind and IMF observations at r (for the kinematic assumption of constant V). The gray histogram in Figure 2 shows the distribution of ΔBr values computed using equations (5) and (6) from hourly observations taken near Earth (r = r1 = 1 AU) between 1963 and 2008 as compiled in the Omni-2 data set of hourly interplanetary averages (the continuation of the work of Couzens and King [1986]).

Figure 2.

The distribution of predicted radial field difference (normalized to r = r1 = 1AU) c × ΔBr1 = c × {[Br]r1 − (r0/r1)2[Br]ro} between r1 = 1AU and the coronal source surface at ro = 2.5/251 AU. The shaded area is for c = 1 (and so includes the effect of the polarity of Bφ), and the black line is for c = Bφ/∣Bφ∣ (i.e., c = ± 1) which removes the effect of the polarity of Bφ. The plot is for hourly data from the entire Omni-2 data set for 1963–2008.

[8] In compression regions where solar wind speeds increase at any one latitude (dV > 0), dynamical stream–stream interaction regions form on the leading boundaries of coronal holes [Cranmer, 2002] where fast wind catches up with slow wind ahead of it and the assumption of constant V becomes invalid. These flow gradients steepen as they propagate outward but do not generally form shocks until r ≈ 2 AU [e.g., Gosling, 1996]. Such interaction regions at r = 1 AU typically last on the order of 1 day [e.g., Gosling and Pizzo, 1999]. To remove their effect on the kinematic correction, we here use smoothed values of V: daily means in dV are interpolated to apply at the centers of the hourly averaging intervals of the IMF data. The smoothing time constant of 1 day represents a compromise: it smoothes out much of the dynamical effects in interaction regions, but the overall autocorrelation function of the solar wind speed at 1 AU falls to 0.5 for lags near 30 h [Lockwood, 2002], hence using intervals any longer than 1 day would cause us to also average out considerable larger-scale longitudinal structure in the source solar wind speed. The use of 1 day smoothing means that we are predicting the kinematic effects of flow speed structure on timescales of 1–27 days. The effectiveness of this procedure to remove dynamic interaction effects in compression regions is discussed later in section 2.

[9] From equation (6), Burlaga and Barouch [1976] show that the parameter μ is given by

equation image

Hence computation of μ as a function of r requires knowledge of the ratio [∂V/∂t]o/V2. This ratio is shown in Figure 3 for the same data set as Figure 2. Positive values are compression regions ([∂V/∂t]o > 0) which give dt < dto, negative values are rarefaction regions ([∂V/∂t]o < 0) which give dt > dto. Note that V values have been given in units of AU s−1 as this gives ready application with r values expressed in units of AU.

Figure 3.

The distribution of predicted values at r = ro of the ratio [∂V/∂t]o/V2 (with V here expressed in units of AU s−1). The ∂V values are interpolated from daily means to remove effects of fine structure in the variation in V. The plot shows the hourly values from the entire Omni-2 data set for 1963–2008. Positive/negative values are for compression/rarefaction regions, respectively.

[10] Studying equations (4) and (7) highlights the limitations of this theory in compression regions. At sufficiently large r, {(rro) [∂V/∂t]o/V2} rises to unity for [∂V/∂t]o > 0 and hence, by (7), μ becomes infinite and, by (4), so does Bφr.

[11] Equation (5) shows that for a fixed r and λ, the ratio of {[∂V/∂t]o/V2}/ΔBr depends on (VBφr). For a positive/negative Bφr, a given polarity of [∂V/∂t]o will give the same/opposite polarity of ΔBr, respectively. To remove the dependence on the polarity of Bφr, we here use c = Bφr/∣Bφr∣ = ± 1. Figure 4 shows the distribution of values of c{[∂V/∂t]o/V2}/ΔBr from the same data set as in Figures 2 and 3. The mode value of 0.02 applies to both compression and rarefaction regions.

Figure 4.

The distribution of predicted values of the factor c{[∂V/∂t]o/V2}/ΔBr (with V expressed in units of AU s−1). The ∂V values are as used in Figure 3. The term c = Bφ/∣Bφ∣ = ± 1 removes the effect of the polarity of Bφ. The plot shows the hourly values from the entire Omni-2 data set for 1963–2008. The mode value is 0.2, and this distribution includes both compression ([∂V/∂t]o > 0, ΔBr/c > 0) and rarefaction ([∂V/∂t]o < 0, ΔBr/c < 0) regions.

[12] As in paper 1, we define the signed flux deficit for measurements made at r and r1 to be

equation image

The radial term of equation (2) gives Br = (ro/r)2Bro + ΔBr and Br1 = (ro/r1)2Bro + ΔBr1. Substituting into (8), using equation (5) and rearranging gives the kinematic contribution to excess flux:

equation image

All of the above equations assume that solar wind speed V does not depend on r but, as discussed above, when dV > 0 dynamical effects become important. This applies to the corotating interaction regions, as well as to temporal cases such as ahead of coronal mass ejections (CMEs). These dynamical effects are absent in rarefaction regions (dV < 0) because the faster flow runs ahead of the slow flow. Hence it is instructive to repeat the distribution shown in gray in Figure 2 for values of cΔBr, for which c removes the effect of the polarity of Bφr: hence cΔBr > 0 corresponds to dV > 0 (compression regions) and cΔBr < 0 corresponds to dV < 0 (rarefaction regions). This is shown by the black line in Figure 2. It can be seen that the symmetric distribution for c = +1 has become slightly asymmetric with slightly lower numbers of samples giving large positive cΔBr than give large negative cΔBr of the same magnitude. This is consistent with the asymmetry expected for the dynamical stream–stream interaction effects which will be present in compression regions only. Thus we can conclude that using 1-day means to derive (dV/dt) has removed much, but not all, dynamical effects in the excess flux calculation. Because measurement uncertainties and our assumptions can sometimes (in <1% of hourly means) cause exceptionally large values of ∣ΔBr∣, we here exclude from the survey all samples which are more than 3σ from the mean (the standard deviation from Figure 2 being σ = 2.24 nT).

[13] Taking the integral of all the 87,759 available hourly positive cΔBr values in Figure 2 up to this 3σ limit (i.e., in the compression regions) we get 0.33 Ts, whereas the corresponding integral value for all the 99,694 hourly negative cΔBr values (i.e., in the rarefaction regions where dV < 0) is −0.40 Ts. Because there are no stream–stream interaction effects in the rarefaction regions, and because the true integrated excess flux effect for dV > 0 regions should be equal and opposite to that for dV < 0: we can infer that the dynamical interactions have caused the procedure described above to underestimate the kinematic effect in compression regions by a factor 100(0.40–0.33)/0.40% = 17.5%, on average. Given the dV > 0 regions are present for 46.8% of the time, we can estimate that we may have underestimated the kinematic excess flux effect by some 0.468 × 17.5 = 8.2%, on average. In theory, we could use Figure 2 to calibrate for this effect and remove the average residual dynamic interaction effects in compression regions: however, we do not attempt to implement such an average correction here and instead note the average level of underestimation of the kinematic effect owing to dynamical effects in compression regions.

3. Kinematic Effects on Sampling

[14] The kinematic effects discussed in section 2 cause a bias in the sampling of the IMF/slow wind seen in the heliosphere [McComas et al., 1992]. Regions of [∂V/∂t]o > 0 give dt < dto, which means that a satellite at r > ro will sample the resulting compression regions relatively more often than at the source region. Conversely regions of [∂V/∂t]o < 0 give dt > dto and so fewer of the regular samples at the satellite will relate to these rarefaction regions, compared to at the source region. Hence if the field at the source surface is systematically different in the regions of [∂V/∂t]o > 0, this would introduce a bias into the averages seen at r > ro.

[15] From equation (6) the value of dto corresponding to the sampling interval dt1 can be calculated for all the Omni-2 data from r = r1, with the kinematic assumption that V is independent of r. In addition to the conventional mean for a solar rotation, 〈∣Br1∣〉CR, we can compute a mean where each sample of ∣Br1∣ is weighted by a factor w = (dto/dt1)/{ΣCR(dto/dt1)}, which means it is weighted by the time interval relevant to at the source surface, rather than that at the point of observation. Figure 5 is a scatterplot of 〈∣Br1∣〉CR and 〈wBr1∣〉CR which shows there is no systematic difference between the two. Circles are averages for data in rarefaction regions (dV < 0) whereas the triangles are for compression regions (dV > 0). Individual solar rotations may show some deviation of 〈wBr1∣〉CR from 〈∣Br1∣〉CR, and there is a slight tendency for the biggest deviations to be in compression regions. The scatter for this plot (which considers the difference between ro = (2.5/251)AU and r1 = 1 AU) is somewhat greater than for the corresponding plot (Figure 5) in paper 1 for between r1 = 1 AU and Ulysses at rU ≈1.5 AU. We believe that this is partly because the range of r is greater and includes the solar wind acceleration region. However, on average, 〈wBr1∣〉CR and 〈∣Br1∣〉CR still agree very closely for both dV > 0 and dV < 0. As in paper 1, we can conclude that kinematic effects have not introduced a bias and this conclusion is not effected by dynamical effects in interaction regions (which are absent in rarefaction regions). Hence we can eliminate the kinematic bunching effect on sampling as a contributor to the flux excess. We have confirmed this conclusion using the theoretical model of kinematic structures developed by A. P. Rouillard and M. Lockwood (Solar stream magnetism: Analytic prediction of three-dimensional heliospheric fields and flows, submitted to Astronomy and Astrophysics, 2009).

Figure 5.

Analysis of the effect of kinematic bunching on the sampling of radial field values. Averages for Bartels rotations are shown for the entire Omni-2 data set for 1963–2008. The abscissa shows the mean radial field seen by near-Earth spacecraft (r = r1 = 1 AU) for sampling intervals dt1 = 1 h, 〈∣Br∣〉. The ordinate gives the mean for the weighted observed radial field values, for a weighting factor of (dto/dt1), where dto is the time separation at r = ro which corresponds to dt1. Triangles are for data in compression regions, dV/dt > 0, giving dto > dt1; circles are for data in rarefaction regions, dV/dt > 0, for which dto < dt1. The raw data means and the weighted means are equal along the diagonal line. It can be seen that the sampling bias introduced by compression/rarefaction during one solar rotation is generally (but not always) very small and that no bias is introduced on average. Almost no difference is seen in average values between compression and rarefaction regions.

4. Kinematic Effect on the Radial Field and Flux Excess

[16] Figure 2 shows the distribution of hourly ΔBr values computed using equation (6) for all the near-Earth (rr1 = 1 AU) observations between 1963 and 2008 in the Omni-2 data set. In order to evaluate the net effect over each solar rotation, we plot the equivalent distributions, an example being given in Figure 6 (top). As discussed in paper 1, it is valuable to use integer 27-day intervals and so we use Bartels rotation intervals rather than Carrington rotations (the former being 27 days the latter 27.2753 days). This particular example is for Bartels rotation number 2226. The number of hourly Omni-2 data points giving a predicted additional radial field of [ΔBr]k is termed nk and so the fraction of the total possible number (N = 27 × 24) of hourly samples available for this particular solar rotation is f = Σ knk/N = 1.0; that is, there are no data gaps in this case. It can be seen that the distribution is quite symmetric (i.e., roughly half the predicted extra flux generated by kinematic effects is outward and half is inward), but the distribution is not as symmetric as the equivalent plot in Figure 2 (the gray area for c = +1) which is for all the data.

Figure 6.

(top) The same as Figure 2 but for a single Bartels rotation (in this example, number 2226). The number of hourly Omni-2 data points giving a predicted additional radial field of [ΔBr1]k is nk. The fraction of the total possible number (N = 27 × 24) of hourly samples available for this particular solar rotation is f = Σ k(nk/N) = 1.0; that is, there are no data gaps in this case. (bottom) The ordinate is [ΔF1]CR/(NA1), where [ΔF1]CR is the magnitude of the integrated additional radial field up to the limit [ΔBr]i (abscissa); i.e., [ΔF1]CR = Σikk = 0A1nk∣ΔBr1k, where A1 is the area of the surface of the heliocentric sphere of radius r1 that is swept out by unit length in the N direction (of the RTN coordinate system) by solar rotation during 1 h. The value at [ΔBr1]i = −10 nT is the predicted total of all detected inward extra flux [ΔF1]in and that at [ΔBr1]i = +10 nT is the corresponding integrated outward flux [ΔF1]out. For stationary conditions over the solar rotation, with purely radial flow and no vestigial net effect of stream–stream interactions, we would expect [ΔF1]in = [ΔF1]out and the mean additional radial field produced by kinematic effects would be ∣ΔBr1CR = 2[ΔF1]in/(NA1) = 2[ΔF1]out/(NA1). The dashed horizontal line shows the average of [ΔF1]in and [ΔF1]out which is here taken to be (NA1) ∣ΔBr1CR/2.

[17] To evaluate the effect of this extra inward/outward flux on the 〈∣Br1∣〉CR and open flux estimates, we here compute the integral from zero to each additional radial field of [ΔBr]i, giving a magnetic flux of [ΔF1]CR = Σikk = 0A1nk∣ΔBrk (where A1 is the area of the surface of the heliocentric sphere of radius r1, that is swept out by unit latitudinal length by solar rotation during the hour). Figure 6 (bottom) shows [ΔF1]CR/(NA1) as a function of [ΔBr]i. The value at [ΔBr]i = −10 nT is the total of all predicted inward extra flux [ΔF1]in during the solar rotation in question and that at [ΔBr]i = +10 nT is the corresponding integrated outward extra flux [ΔF1]out. In this case [ΔF1]in is slightly smaller than [ΔF1]out: for some other solar rotations it is the other way round and only relatively rarely are the two exactly equal. For stationary conditions over the solar rotation, with purely radial flow and no residual contributions of dynamic effects in compression regions, we would expect [ΔF1]in = [ΔF1]out and the mean additional radial field produced by kinematic effects would then be ∣ΔBrCR = 2[ΔF1]in/(NA1) = 2[ΔF1]out/(NA1), the factor of 2 arising because both the inward and outward flux contribute to the absolute value of the radial field. We attribute the differences between [ΔF1]in and [ΔF1]out detected to either nonstationary conditions over the solar rotation, to residual dynamical effects, to data gaps and/or to nonradial flow. We here take the arithmetic mean of [ΔF1]in and [ΔF1]out (the dashed horizontal line in Figure 6 (bottom)) to be equal to (NA1) ∣ΔBr1CR/2. Using this estimate of ∣ΔBr1CR will introduce some scatter into the values for individual solar rotations, but which should average out when sufficient solar rotations are considered together (e.g., in annual means).

[18] Figure 7 shows solar rotation averages of the observed flux excess ΔFS, computed as a function of r using equation (8), from a survey of data from a number of spacecraft throughout the heliosphere. This is an updated version of Owens et al. [2008a, Figure 1]. Points are colored according to the spacecraft giving the observations away from 1 AU, as given by the key. The simultaneous data from 1 AU is taken from the Omni-2 data set. It can be seen the excess flux increases with r as does the scatter about the trend.

Figure 7.

The observed additional open flux ΔFS as a function of r/r1. The colored dots are the differences between full solar rotation averages observations from various spacecraft, compared with the coincident Omni-2 data value. The lines are from the predicted ∣ΔBr1CR from the Omni-2 data using the mode value of the distribution shown in Figure 3 (see text). The lines show the ΔFS values which would be exceeded a fraction p of the time where p is 0.1 (for black line), 0.25, 0.5, 0.75 and 0.9 (light gray line).

[19] The lines in Figure 7 are predicted using the theory given in section 2 in the following manner. The values of ∣ΔBr1CR for each solar rotation are computed (as demonstrated by Figure 6) and from the distribution of these solar rotation values are taken the upper and lower deciles, the upper and lower quartiles and the median (i.e., the probability of ∣ΔBr1CR exceeding these values is p equal to 0.9, 0.1, 0.75, 0.25 and 0.5). These five values are then used to generate the corresponding ∣∂V/∂t∣o/V2 values using the mode value of the distribution shown in Figure 4 (which is m = 0.2 AU−1 nT−1). These are then used to generate a pair of μ(r) profiles for each of the 5 p values using equation (7) with [∂V/∂t]o/V2 = ± m ∣ΔBr1CR (the plus being for compression regions, the minus being for rarefaction regions). These are then averaged together and the average μ(r) profile is then used with equation (9) to extrapolate each of the above five values to other r. Extrapolation has been curtailed at the point where the compression regions are tending toward giving infinite field as this is where the kinematic disturbance would steepen into a shock [Burlaga et al., 1983].

[20] Thus we can predict the flux excess ΔFS that we would expect at a given r to be exceeded with a probability p of 0.1, 0.25, 0.5, 0.75 and 0.9 of the time. These are the five lines in Figure 7 (shaded from black to light gray). It can be seen that these fit the observations reasonably well and shows that the flux excess is indeed consistent with the kinematic effects described in section 2. The spread of the lines for different p matches the observed scatter reasonably well.

[21] Figure 8 shows the data from Figure 7 for r < 5 AU. Instead of individual solar rotation values of ΔFS, means for bins of r that are 0.1 AU wide are shown with error bars of plus and minus one standard deviation. Figure 8 (bottom) shows the number of solar rotation averages in each bin (on a logarithmic scale). The five lines are the same as in Figure 7. By definition of ΔFS, all five lines pass through zero at r = 1 AU. In addition to the satellite data from Figure 7, the mean and standard deviation of all PFSS values are shown by the leftmost data point in Figure 8. It can be seen that the flux excesses for these PFSS estimates lie on the same trend with r as do the satellite data (and are negative; i.e., the PFSS values are lower than those from r = 1 AU). We therefore infer that the tendency for lower PFSS values noted by Wang and Sheeley [1995] and Lockwood et al. [2006] has the same origin as those noted in spacecraft data by Owens et al. [2008a] and both are consistent with kinematic effects introduced by large-scale spatial velocity gradients.

Figure 8.

(top) The additional open flux ΔFS as a function of r/r1, where r is the heliocentric distance and r1 = 1 AU. The data points show the mean values from the spacecraft data shown in Figure 7, averaged over r bins 0.1 AU wide, with error bars of plus and minus one standard deviation. The lines are the same as shown in Figure 7. The data point at the lowest r/r1 is the entire PFSS data set for the coronal source surface at r = ro. (bottom) The number of full solar rotation averages n contributing to the means. The open triangle shows the number in the Omni-2 data set from r = r1. Note that n is shown on a logarithmic scale.

5. Long-Term Variations in Open Solar Flux

[22] Figure 9 shows annual means of the open flux derived for averaging timescales T of 1 h, 1 day, 2 days and 3 days (gray shaded areas). The red line shows the variation for T = 1 h, minus the kinematic correction ∣ΔBr1CR derived as described above, [FS]C = 2πr12{〈[Br]T = 1hrCR − 〈ΔBr1CR}. Note that because this correction uses the observed tangential field value Bφr1 at r = r1 with equation (5), it does not rely on the average procedure used to obtain μ(r) and hence generate the ΔFS(r) profiles in Figures 7 and 8. The green line shows the values derived from magnetograph data using the PFSS method [Wang and Sheeley, 1995], [FS]PFSS. Additional data gaps appear early in the corrected data sequence (in red) because we require both IMF and solar wind data and we set a requirement 50% data availability. It can be seen that the corrected open flux values match the PFSS data rather well, better than those obtained using T of 1–3 days. The r.m.s. difference between the PFSS values and the kinematically corrected open flux values is 4.1 × 1013 Wb, whereas the corresponding r.m.s. difference for using averaging over T = 1 day is 6.9 × 1013 Wb. Thus the kinematic correction performs better on average but not dramatically so.

Figure 9.

The gray shaded areas show annual means of the signed open solar flux from the Omni-2 data [FS]T = 2πr12 〈∣Br1T〉, with absolute values taken of means on timescale T. The light gray area bounded by the blue line is for T = 1 h ([FS]T = 1hr) and successively darker gray areas are for T = 1, 2, and 3 days. The green line is the corresponding value from the PFSS data, mapped to r = r1, with no allowance for kinematic effects, [FS]PFSS. The red line shows the Omni-2 values for T = 1 h, minus the correction term for kinematic effects, [FS]C = 2πr12 {〈∣BrT = 1hrCR − 〈ΔBr1CR}.

[23] Last, Figure 10 shows the variation of solar rotation averages (thin dark gray line) and annual means (thick black line) of the open solar flux, corrected for the kinematic effects, as described above. The most notable feature is how rapid the descent of open solar flux has been over recent cycles. The annual mean open flux for 1987 is a solar minimum value of 0.382 × 1015 Wb, whereas the annual mean open flux in 2007, also a solar minimum value, is 0.198 × 1015 Wb. In other words, the annual mean open solar flux was 93% higher in 1987 than it was in 2007.

Figure 10.

The signed open solar flux, corrected for the excess flux using the kinematic effect discussed in this paper, [FS]C = 〈[FS]T = 1hrCR − 2πr12∣ΔBrCR. Annual means are shown by the thick black line (also shown by the red line in Figure 9); averages over Bartels rotations are shown by the thin gray line.

6. Discussion and Conclusions

[24] The analysis presented here has shown that the flux excess seen by various spacecraft in the inner heliosphere (mainly close to the ecliptic plane) is consistent with the difference between values derived using in situ data from heliocentric distances of r ≈ 1 AU compared to those deduced from solar magnetograph data using the PFSS method. In addition, we have shown both are consistent with kinematic effects in the streamer belt owing to large-scale (timescales T > 1 day) longitudinal structure in the solar wind flow.

[25] There is no kinematic effect unless there is a tangential seed field in the region of high solar wind speed shear. The origin of such tangential seed fields could be, for example, near-Sun interchange reconnection of magnetic field or could be faster flow emerging beneath overdraped Parker spiral field (temporal effects such as CMEs). Because the analysis starts from the observed field at r near 1 AU and maps back to the source surface (r = ro), our results do not depend on the mechanism which generates the tangential seed field in the first instance. We have investigated the effect of the biggest assumption made (that V does not depend on r; i.e., assuming that dynamical stream–stream interaction effects have been largely removed by the 1-day smoothing time constant applied to the observed velocity gradient) and found that the resulting difference between compression and rarefaction regions is small. We conclude the vestigial effects of stream–stream interactions in our kinematic correction terms are small and tend to cancel out.

[26] Figure 9 shows that the temporal variation of the annual mean of the corrected open flux [FS]C, derived by applying the kinematic correction for excess flux to whole-solar rotation data from spacecraft near 1 AU, matches very well that from PFSS modeling on the basis of magnetograph data. In particular, the two agree somewhat better than applying by preaveraging over a fixed interval T. Up until about 1986 (solar cycles 20 and 21), using T = 1 day ([FS]T = 1 day) matches the PFSS data rather well; however, over cycle 22 [FS]T = 2 day matches better and over cycle 23 even [FS]T = 3 day does not give an adequate correction. As discussed in the introduction, there will, at any one time, be a T value for which the derived open flux at 1 AU equals the correct value at the source surface; however, Figure 9 implies that the optimum T at 1 AU has changed. The agreement with PFSS open flux estimates is improved somewhat if we use the kinematic correction rather than averaging over intervals of T = 1 day (the r.m.s. deviation is reduced by about a third).

[27] Paper 1 showed that small-scale structure (timescales less than 1 h) does not contribute to flux excess, but structure on timescales between 1 h and 1 day could still be a factor: the use of means over, for example, T = 1 day would be the best way to remove such structure. Because of dynamical interactions, our analysis of kinematic effects needs to smooth the observed velocity gradients (we use a time constant of 1 day) and so we here have predicted the kinematic effects of flow structure on timescales of >1 day. These can match the observed flux excess, and its variations with r and time, rather well. However, we cannot eliminate the possibility that there is a contribution to the excess flux from structure on timescales between 1 h and 1 day (at least some of which could come from dynamical effects). In this discussion about the physical origin of the flux excess, it is interesting to note the latitudinal difference reported in paper 1: that the flux excess was considerably greater in the streamer belt than outside it. Lockwood and Owens [2009] show that, in fact, this only applies around sunspot minimum and that near sunspot maximum the flux excess is roughly as great at high heliographic latitudes as at low latitudes. This clearly points to kinematic and/or dynamical interaction effects as the physical origin of the flux excess. Naturally, in general, one cannot have one without the other. However, here we have used smoothing to damp the dynamical effects (and shown that this is approximately achieved because rarefaction and compression show similar results) and find a close agreement of the data with the kinematic effects predicted for the background the solar wind flow structure on timescales of >1 day. However, this does not eliminate contributions to the excess flux of structure in the field from another source on timescales in the range 1 h to 1 day.

[28] Figure 9 shows that the open solar flux in the current solar minimum is lower than at any previous time since measurements of interplanetary space began in 1963, and that rapid descent has occurred since the maximum of the long-term variation in open solar flux in 1987 identified by Lockwood [2001, 2003] and Lockwood and Fröhlich [2007]. Here we note, the correction 2πr12∣ΔBr1CR (needed to match PFSS values) has increased slightly in magnitude. Note that the low values of open fluxes for the current solar minimum do not only originate from data from the ACE and WIND spacecraft (which are the major contributors to the Omni-2 data at this time): Paper 1 shows that they are consistent with the Ulysses data at larger r and Owens et al. [2008a] have shown that values from the two STEREO craft, at similar r but different solar longitude ϕ, are almost identical.

[29] A number of models for the evolution of the heliospheric magnetic field have been proposed. Fisk et al. [1999] argue that the Sun's open flux tends to be conserved, with “interchange reconnection” [see Crooker et al., 2002] between open and closed solar fields resulting in an effective diffusion of open flux across the solar surface without, necessarily, any net change in the total open flux. In this case, the heliospheric field evolves with simple rotation of regions of positive and negative polarity separated by a single, large-scale heliospheric current sheet [Fisk and Schwadron, 2001; Jones et al., 2003]. Alternatively, it has been argued that emerging midlatitude bipoles cause closed coronal loops to rise and first destroy preexisting open flux in the polar coronal hole (remnant from the previous solar cycle) and then build up a new polar coronal hole (of the opposite polarity) and so reverse the polar field of the Sun [Babcock, 1961; Wang and Sheeley, 2003], which fits well with the migration of photospheric fields inferred from magnetograph data. The evolution of the heliospheric magnetic field could also be facilitated by transient events [Low, 2001]: specifically, Owens et al. [2007] and Owens and Crooker [2006, 2007] investigate the role of the magnetic flux contained in coronal mass ejections (CMEs) in the observed variation in flux seen by craft in the heliosphere. These different concepts are not mutually exclusive in many respects (see review by Lockwood [2004]).

[30] Much of the difference between these concepts is a matter of semantics. “Open flux” has here, and in many previous papers, been taken to be the same as “coronal source flux;” that is, the magnetic flux that leaves the solar atmosphere and enters the heliosphere by threading the coronal source surface at 2.5 solar radii. It is a readily measurable quantity because of PFSS modeling (within the assumptions of that technique) and because the Ulysses result allows the use of in situ magnetic field data (but we have shown that some form of correction is needed for the excess flux effect). This is quite different from another definition of open flux which requires that it has only one foot point still attached to the Sun [e.g., Schwadron et al., 2008]. Flux which appears to be in this category can sometimes be inferred for in situ point measurements, for example from heat flux or unidirectional “strahl” electron distribution functions, although scattering by heliospheric structure into other populations such as “halo” often makes this far from unambiguous [Larson et al., 1997; Fitzenreiter et al., 1998; Owens et al., 2008c]. Even if this could be done reliably, there is no way to quantify the total of such flux at any one time from such in situ point measurements. This is because there is no equivalent of the Ulysses result (and so no equivalent to equation (1) for this flux) to generalize in situ point measurements into a global quantity. To make the distinction clear, let us here refer to these two definitions as the “total open flux” FS (defined by equation (1)) and the “single–foot point open flux” FO. The latter is a subset of the former. The only topological distinction that FO can have which separates it from other heliospheric flux (the FSFO of double–foot point open flux) is that it threads the heliopause and enters interstellar space. (Note that any other definition involving any boundary within the heliosphere will cause the continuous conversion of the flux (FSFO) into FO as the magnetic flux frozen-in to the solar wind flow propagates through that boundary).

[31] Until we can quantify FO, there can be no evidence that it is constant and thus the idea that it is constant can be no more than a hypothesis. Confusingly, it has been claimed that the total open flux FS is constant and that this is evidence that the single–foot point open flux FO is constant. The key point we wish to underline here is that FS is far from constant.

[32] Coronal mass ejections are an important example (but not the only example) of flux that undoubtedly contributes to FS but may not contribute to FO [Mackay and van Ballegooijen, 2006]. Consider a magnetic flux tube which is an element of a CME and which contains a magnetic flux FE. It will enhance the flux FS by 2FE (FE at each foot point), once it has propagated beyond the coronal source surface without any foot point disconnection. This emergence does not change FO. However, unless FS is to increase indefinitely with continuing CME emergence, other processes must occur. Emergence followed by subsequent reconnection with a preexisting similar loop (i.e., with dual–foot point open flux which is part of FS but not part of FO) will give no net change in either FS or FO. Emergence with subsequent disconnection in one hemisphere A by magnetic reconnection with preexisting single–foot point open field (residual from the previous solar cycle and thus with the opposite polarity) will also give no net change in neither FS nor FO. (Note that single–foot point flux of the old polarity attached to hemisphere A has decreased by FE whereas the new polarity flux attached to hemisphere B has increased by the same amount; i.e., this is foot point exchange). If reconnection of CME flux with single–foot point open flux can occur in hemisphere A, there is no reason why sometimes it cannot also occur in the other hemisphere B (either simultaneously or, more likely, sometime before/after). Emergence followed by complete foot point disconnection (in both hemispheres) causes both FS and FO to fall by 2FE. The CME processes therefore acts to either conserve or decrease FO. If FO is decreased, the counterbalancing source would be reconnection at the heliopause of dual–foot point open flux (FSFO) with the magnetic field in interstellar space (which increases FO but does not alter FS). This would be, at most, only distantly related to the near-Sun processes.

[33] The time series of the total open flux FS shown in this paper (Figure 10) and by prior publications reveals that there is considerable variation. The total open solar flux at solar minimum fell from an annual mean of 3.82 × 1016 Wb in 1987 to close to half that value (1.98 × 1016 Wb) in 2008.

[34] Long-term variations in the total open solar flux FS have previously been inferred from historic geomagnetic data by Lockwood et al. [1999a, 1999b], Lockwood [2001, 2003] and Rouillard et al. [2007], showing that the open solar flux roughly doubled between 1900 and about 1950. It has been claimed that this rise was an artifact of the aa geomagnetic data [Svalgaard et al., 2003, 2004] or was present but much smaller in magnitude [Svalgaard and Cliver, 2007]. Analysis using a wide variety geomagnetic data show neither to be the case [e.g., Rouillard et al., 2007]. In this debate, the important difference between open solar flux FS (and hence by equation 1 the radial field component) and the heliospheric field strength B has also often been overlooked. As a result, discussion of the existence [Svalgaard and Cliver, 2005], or otherwise [Owens et al., 2008b], of a “floor” minimum to B is irrelevant. Rouillard et al. [2007] pointed out the role of even uniform solar wind flow speed V in decoupling FS and B. (In Parker spiral theory, increased/decreased V causes the spiral to unwind/wind up and so B falls/rises for a fixed FS). But these authors point out this is not the only effect which in the past has been accounted for using the timescale T [Rouillard et al., 2007; Lockwood et al., 2006]. In the present paper, we have shown that this additional effect is consistent with longitudinal structure in the solar wind flow which gives the kinematic flux excess effect.

[35] The long-term change in the open flux deduced from geomagnetic activity has been reproduced by a number of numerical models of flux continuity and transport during the solar magnetic cycle, given the variation in photospheric emergence rate indicated by sunspot numbers [Solanki et al., 2000, 2002; Schrijver et al., 2002; Lean et al., 2002; Wang and Sheeley, 2003; Wang et al., 2005]. The principle laid down by Solanki et al. [2000, 2002] is that total open flux FS obeys a continuity equation, with the rate of change being the difference between a source terms (the total rate that coronal field loops emerge through the coronal source surface, including CMEs) and loss terms owing to field reconfiguration and disconnection by magnetic reconnection. The discussion above about CME effects illustrates that several different topologies of disconnection must be active (see review by Lockwood [2004]). In the absence of known mechanisms that could make the total loss (from the variety of mechanisms) exactly equal to the simultaneous production rate, we must expect the total open flux to vary on a variety of timescales.

[36] In this context, it is worth noting that the annual mean solar minimum open solar flux, derived here with correction for kinematic effects, was almost twice as large in 1987 than it was in 2007. This means that the observed fall in the minimum value over the last two solar cycles was considerably faster than the rise inferred from geomagnetic activity observations over four solar cycles in the first half of the 20th century.


[37] All three authors are supported by the UK Science and Technology Facilities Council. We are also grateful to the Space Physics Data Facility and the National Space Science Data Center for provision of the Omni-2 data set and to many scientists who contributed to both Omni-2 and to other magnetic field observations used here: from Pioneer 6 and 7 (principal investigator N. Ness), Pioneer 10 and 11 (principal investigator E. Smith), Pioneer Venus Orbiter (principal investigator C. Russell), Helios (principal investigator N. Ness), Voyager (principal investigator L. Burlaga), ICE-ISEE3 (principal investigator E. Smith), Ulysses (principal investigator A. Balogh), and STEREO (principal investigator M. Acuña), and to the Small Bodies Node of the Planetary Data System for NEAR magnetometer data (principal investigator M. Acuña). We also thank Yi-Ming Wang for the provision of the PFSS data.

[38] Amitava Bhattacharjee thanks the reviewers for their assistance in evaluating this paper.