SBW is now calibrated in terms of solar wind speed, V SW , using a combination of in situ spacecraft observations and the “Magnetohydrodynamics Around a Sphere” (MAS32) global coronal model constrained by photospheric magnetic field observations. For a given Carrington rotation (CR), the MAS model extrapolates the photospheric field distribution outward to 30 solar radii (R S ), while self-consistently solving the plasma parameters on a non-uniform grid in polar coordinates, using the MHD equations and the vector potential A (where the magnetic field, B, is given by ∇ × A), such that ∇.∇× A = 0 (which ensures current continuity, ∇. J = 0, is conserved to within the model’s numerical accuracy). All model data can be downloaded from http://www.predsci.com/mhdweb/home.php. The calibration period covers 1975–2013, the timespan for which both photospheric magnetograms (and hence MAS estimates) and SBW estimates are available. A combination of Kitt Peak, Wilcox Solar Observatory, Mount Wilson Solar Observatory, SOLIS and GONG data are used to minimse data gaps and provide the longest possible time sequence. V SW is extracted at 30 R S , the interface of the coronal and solar wind models.

Comparison of MAS with spacecraft data reveals MAS reproduces the observed solar wind structure, but generally underestimates the fast wind speed. A linear scaling of 1.04 and an offset of 78.5 km/s is determined to provide the best fit of MAS 27-day means (discussed below) to spacecraft V SW observations. This corrected MAS solar wind speed is referred to as V MAS and shown in Fig. 2. To later enable comparison with the annual sunspot-based SBW estimates, which provide information about the latitudinal solar wind structure, but no longitudinal information, “zonal means” (i.e., longitudinal or 27-day averages) are used. Data are further smoothed using a 1-year rolling (boxcar) means, to mimic the annual SBW estimates while retaining rapid latitudinal variations of the observing spacecraft.

Figure 2 (a) The black-shaded region and left-hand axis show monthly sunspot number over the period 1975–2013. The right-hand axis, and red and white curves show the heliolatitude of the Ulysses and OMNI near-Earth spacecraft, respectively. (b) 1-year rolling means of solar wind speed in near-Earth space observed by the OMNI spacecraft (black), predicted by the MAS model (V MAS , blue) and the sunspot-based reconstruction (V RECON , red). (c) The same as panel (b), but for the Ulysses spacecraft. Full size image

Figure 2a shows sunspot number and the latitude of the observing spacecraft, with the near-Earth OMNI data11 in white and Ulysses33 in red. Figure 2b shows in-ecliptic V SW observations from OMNI (black) and V MAS (blue) at Earth’s heliographic latitude. There is only a very weak solar-cycle variation annual V SW . Indeed, to first order, the OMNI V SW can be approximated as 430 km/s with ~50 km/s variability. In that respect, V MAS is in good agreement. Observations made by the Ulysses spacecraft, shown in Fig. 2c, provide a better picture of global solar wind structure. Ulysses performed 3 “fast-latitude scans” of the Sun, sampling all solar latitude within approximately 1 year. For the two solar minimum fast-latitude scans, in approximately 1994–1996 and 2006–2008, the global solar wind structure is fast wind from the poles and a narrow slow wind band at the equator. This is well reproduced by MAS. The broader slow wind band during the second solar minimum pass (2006–2008), partly the result of increased pseudostreamer occurrence2, is also captured by MAS. During the solar maximum fast-latitude scan, 2000–2002, the agreement is good in the northern hemisphere, but V MAS is higher than observed in the southern hemisphere34. In general, however, there is sufficient agreement between V MAS and observed V SW that we proceed in using V MAS to calibrate the sunspot-based reconstructions.

An example of V MAS for a single Carrington rotation (CR1756, which approximately spans December 1984, the late declining phase of solar cycle 21), is shown in Fig. 3. Panel a shows V MAS at 30 R S in heliographic coordinates. The data have been reduced in resolution to 90 and 45 equally-spaced grid points in longitude and sine latitude, respectively, to enable efficient manipulation of the data. The heliospheric current sheet (HCS) is shown as the thin white line. The dashed black line in panel b shows the zonal mean V MAS for CR1756. As is typical of non-solar maximum periods, the solar wind structure can be approximated as fast wind from the poles, with a roughly equatorial “belt” of slow wind, primarily centred on the HCS. Thus the zonal mean V SW in heliographic coordinates is a product of a number of factors:

1 The inclination of the slow wind belt to the solar rotation direction. Given the association of slow wind with the HCS, this is to first order equal to the magnetic dipole tilt. 2 The angular width of the slow solar wind belt about the HCS location. 3 The higher order “waviness” of the belt. The waviness of the HCS results from quadrapolar (and higher) order components of the magnetic field. 4 Sources of slow wind not directly associated with the HCS (e.g. pseudostreamers or “S-web”35).

Figure 3 A summary of the solar wind speed from the MAS model, V MAS , for Carrington rotation 1756 (December 1984). Panels a and b show V MAS in heliographic coordinates, whereas panels c and d show V MAS inclined to maximise the difference in speed between the poles and equator. Panels a and c show latitude-longitude maps of V MAS at 30 Rs. The HCS is shown as a thin white line. Panels b and d show zonal means of V MAS as a black dashed line. The solid black line shows an average across north and south hemispheres. The red line shows the best functional fit, as described in the text. Full size image

As is shown below, factor 1, the large-scale inclination of the solar wind speed structure to the solar equator, is a strong function of solar cycle phase and thus does not need to be calibrated in terms of SBW. Removal of inclination from V MAS estimates allows the true slow wind band width to be more readily estimated from zonal means. To this end, the V MAS solution at 30 R S is put through a coordinate transform to remove the large-scale inclination of the slow wind band. As will be shown, this is approximately equivalent to transforming to heliomagnetic coordinates. Specifically, we determine the coordinate system which maximises the difference in zonal-mean V MAS between the equator and poles. For CR1756, this requires inclining the North rotation pole up by 32 degrees (through a longitude of −74 degrees). V MAS in this inclined coordinate system is shown in panel c. While the coordinate transform is determined purely from V MAS data, it can clearly be seen that it also reduces the overall magnetic dipole tilt, producing a HCS much closer to the equator (though shorter-scale corrugation is still present). Zonal mean V MAS in the inclined coordinate system (panel d) exhibits a narrower and deeper slow wind belt than in heliographic coordinates. Note that angular width of the zonal mean slow wind band in inclined coordinates still includes the waviness of the slow wind belt (factor 3) and the non-HCS sources (factor 4), which we assume can be calibrated in terms of SBW.

Figure 4 shows zonal mean V MAS for all Carrington rotations in the period 1975–2013. Panel a shows data in heliographic coordinates, while panel c shows the inclined coordinate system, as described above. The angle required for the axis inclination is shown in panel b. As expected, the time variation of the inclination angle approximately follows the HCS tilt (e.g., Fig. 4 of Owens and Lockwood36) with small values at solar minimum, when the rotation and magnetic axes are aligned, and a saw-tooth increase peaking at solar maximum. As can be seen from panel c, inclined coordinates do generally result in a narrower slow wind band than heliographic coordinates. In particular, slow solar wind at mid-latitudes in heliographic coordinates during the declining phase of the solar cycle (e.g., 2002–2004) is largely absent in inclined coordinates, indicating that the broadened zonal-mean slow wind band is the result of increased tilt of the slow wind band to the solar rotation direction, not a broadening of the slow wind band itself. On the other hand, the increased latitudinal extent of the slow wind during the 2007–2009 solar minimum over with the previous 2 minima appears to be the result of both increased tilt and broadening of the band itself.

Figure 4 Summary of V MAS over Carrington rotations 1625 to 2168 (i.e., Feb 1975 to Sept 2015). Panel (a) shows zonal means of V MAS as a function of heliographic latitude and time. Panel (b) show the angular inclination required to maximise the difference in zonal mean V MAS between the poles and equator for each Carrington rotation (black circles). The red line shows an annual resolution mean over all three solar cycles. Panel (c) shows the same as panel (a), but in the inclined coordinate system. Full size image

The next step is to characterise zonal mean V MAS structure for each CR by a reduced number of parameters. We use a simple functional form for the zonal mean V MAS , describing it by a maximum solar wind speed (V 0 ) with a sinusoidal dip, centred on the equator, of depth dV and angular width θ V 37. As SBW contains no hemispheric information, hemispheric averages of zonal V MAS are used, shown as solid black lines in Fig. 3b and d. V 0 is taken to be the maximum value of the hemispheric-averaged zonal mean V MAS . dV is the difference between V 0 and the minimum value of the hemispheric-averaged zonal mean V MAS . θ V is fit as a free parameter. Examples of best fits are shown as the red lines in Fig. 3b and d. In heliographic coordinates, CR1756 yields V 0 = 757 km/s, θ V = 61.1 degrees and dV = 318 km/s. In inclined coordinates it yields V 0 = 757 km/s, θ V = 34.0 degrees and dV = 373 km/s.

The black lines in panels a to c of Fig. 5 show time series of annual means of V 0 , θ V and dV (from data in inclined coordinates) over the period 1975–2013. In panel a, θ V exhibits a strong, saw-tooth-like, solar cycle variation, with an increase in θ V over the last three minima. The 1980 and 2000 maxima show a slower decline in θ V than the 1990 maximum. V 0 is constant at 757 km/s for the bulk of the interval, with short-duration (1–2 years) drops at solar maximum (panel b). This is a result of the prevalence of slow solar wind at all latitudes at solar maximum, both in heliographic and inclined coordinates. dV is only weakly ordered by the solar cycle (panel c).

Figure 5 Panels (a) to (c) show time series of the angular width of slow wind band (θ W ), the maximum solar wind speed (V 0 ) and the depth of the slow wind band (dV), respectively, for MAS solutions (black) and reconstructed from sunspot-based estimates of streamer belt width (red). Panels (d) to (f) show scatter plots of these parameters from MAS with the sunspot-based estimates of streamer belt width (SBW). Red lines show the best fits, described in the text. Full size image