1 Introduction

The solar wind propagates almost radially throughout the heliosphere, meaning the large-scale structure of the solar wind at 1 AU is determined primarily by conditions in the upper corona (see Owens, 2020, and references therein, for a recent overview). In the heliosphere, solar rotation can introduce fast and slow solar-wind streams along the same radial line, leading to the formation of compression and rarefaction regions (Pizzo, 1978). If the coronal structure is stable on time scales comparable to the solar-rotation period, these stream-interaction regions (SIRs) will corotate with the Sun (Wilcox and Ness, 1965; Breen et al., 1998).

Transient structures resulting from coronal mass ejections (CMEs, e.g. Webb and Howard, 2012) propagate through the ‘ambient’ solar-wind structures. While interplanetary CMEs (ICMEs) are responsible for the most severe space weather (Gosling, 1993; Richardson, Cane, and Cliver, 2002), correctly forecasting the ambient solar-wind structure is of space-weather importance for three reasons. Firstly, the ambient solar wind, particularly SIRs, can be geoeffective in its own right (e.g. Richardson, Cane, and Cliver, 2002; Kilpua et al., 2017). Secondly, the ambient solar wind can modulate both the arrival time and properties of ICMEs in near-Earth space (Vrsnak and Gopalswamy, 2002; Cargill, 2004; Case et al., 2008). Thirdly, the ambient solar-wind structure is important for magnetic connectivity between the Sun and Earth, which determines the transport of solar energetic particles (Luhmann et al., 2010; Chollet and Giacalone, 2011).

The magnetic and solar-wind flow structure of the global corona cannot be observed directly, at least not on a routine basis (Antonucci et al., 2020). Instead, the coronal magnetic field (and hence solar-wind flow) can be extrapolated from the observed photospheric field, subject to a number of limitations and approximations (Mackay and Yeates, 2012). Crucially, only the photospheric field on the Earth-pointing face of the Sun can currently be measured, so observations must be accumulated over a full solar rotation (approximately 27.27 days from Earth’s view) to give complete longitudinal coverage (e.g. Hoeksema and Scherrer, 1986). Partly for this reason, coronal models are primarily used in the steady-state approximation, particularly in operational space-weather forecasting. The widely used potential-field source-surface (PFSS. Altschuler and Newkirk, 1969; Schatten, Wilcox, and Ness, 1969; Arge et al., 2003) model is intrinsically steady state, whilst time-dependent magnetohydrodyanmic (MHD) approaches (e.g. Linker et al., 1999; Toth et al., 2005; DeVore and Antiochos, 2008; Yalim, Pogorelov, and Liu, 2017) are run until a steady-sate equilibrium is reached (e.g. Riley et al., 2006). These coronal solutions are then used to drive heliospheric models out to Earth orbit (Riley, Linker, and Mikic, 2001; Odstrcil, 2003; Merkin et al., 2016; Narechania et al., 2021). Methods to produce physically consistent photospheric magnetic-field conditions for the unobserved side of the Sun have been developed (Hickmann et al., 2015), which potentially allow relaxation of the steady-state approximation. However, these time-evolving photospheric fields are still primarily used as ‘snapshots’ with steady-state coronal models. There are on-going efforts to move towards fully dynamic coronal reconstructions, though these are not yet in routine operational forecasting use (Yeates and Mackay, 2009; Yeates et al., 2010). Given the prevalence of steady-state coronal – and hence steady-state solar-wind – models and forecasts, it is useful to quantify the validity of this approximation and how that changes over the solar cycle.

The general expectation is that the corona evolves more rapidly at solar maximum than minimum. Figure 1 shows examples of white-light images of the solar corona from the Large-angle Spectroscopic Coronagraph (LASCO) instrument on board the Solar and Heliospheric Observatory (SOHO) spacecraft (Brueckner et al., 1995). Panel b shows the corona 27 days after panel a. These images were taken in mid-2019, close to solar minimum. The streamers, associated with the slow solar wind, are confined to the equatorial regions in both cases and there does not appear to be substantial evolution between the two time periods. Panels c and d show the same for 2014, close to solar maximum. Streamers are present at all latitudes in both images, but the locations and widths have changed substantially over the 27-day period.

Figure 1
figure 1

White-light images of the solar corona from the Large-angle Spectroscopic Coronagraph (LASCO) instrument on board the SOHO spacecraft. Panels a and b show images 27 days apart close to solar minimum, while panels c and d show images 27 days apart close to solar maximum.

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The rate of change of solar-wind structure is also important for a second aspect of space-weather forecasting. Assimilation of in situ observations into solar-wind models (Lang et al., 2017; Lang and Owens, 2019) has been shown to provide a significant improvement to forecast accuracy (Lang et al., 2021). Remote-sensing observations, such as interplanetary scintillation (Jackson et al., 2015) and, in principle, information from white-light heliospheric imager observations (Barnard et al., 2019, 2020) can also be used. This data assimilation (DA) process requires an estimate of the observation uncertainty. A small part of this uncertainty is the measurement error, but more significant is the representivity error. This quantifies how well the observations match up with the properties represented by the model.

For in situ observations, any latitudinal offset, \(\Delta \theta \), between the point of observation and the latitude being forecast will introduce a spatial observational error (Owens et al., 2020). There is also a temporal aspect to the observation error, best understood through an example. A spacecraft at the L5 Lagrange point, approximately 60∘ behind Earth in its orbit, would observe the solar-wind conditions expected to arrive at Earth in approximately 5 days time (Simunac et al., 2009; Thomas et al., 2018), assuming no time evolution of the coronal solar-wind sources and hence 1-AU structure (this also ignores the \(\Delta \theta \) effect due to the inclination of the ecliptic plane to the solar equator, Owens et al., 2019). Both the near-Earth and the L5 observations contain useful information about the solar-wind structure and hence assimilating these data should (and, in general, does) improve forecasting (Lang et al., 2021). Let us suppose that we want to forecast near-Earth solar-wind conditions three days ahead. Earth’s Carrington longitude in three days time will be the same as that observed at L5 two days previously, and as that last observed at Earth \(27.27 + 3 = 30.27\) days previously. Intuitively, we expect that the newer observation will be of greater value to the DA than the older observation. Hence, a naive approach would be to only use the most recent observation. More desirable would be to use both observations, but to quantitatively weight the observations by their relative merit. At present, however, these weightings are unknown. In particular, it is not clear whether newer observations should be more valued at solar minimum than at maximum, when the corona and hence solar-wind structure is known to be more dynamic. Studies of observation impact within DA can shed light on which observations influence the quality of the forecast (Fowler and van Leeuwen, 2013; Langland and Baker, 2004). However, these efforts in their infancy for solar-wind DA.

This study aims to quantify the rate of change in solar-wind structure at 1 AU using two key parameters: the solar-wind speed, \(V\), and the radial magnetic-field component, \(B_{R}\). These two parameters are chosen as they characterise the ambient solar wind well, representing both the plasma flow and the large-scale heliospheric magnetic-field structure. They are also the parameters that are passed from coronal to heliospheric models and the most critical parameters for solar-wind DA. Section 2 uses in situ solar-wind observations, while Section 3 uses magnetogram-constrained coronal and heliospheric simulations. As properties at 1 AU are directly (though not linearly) related to properties at the top of the corona, the results for 1-AU conditions are expected to be representative of the coronal variation (and vice versa). Analysis is performed at 1 AU for two reasons. First, there is a long record of in situ observations at this location, which enables a ‘sanity check’ on the results obtained from simulations. Secondly, for DA purposes, we require the variability to be quantified at the observation distance of 1 AU.

2 In Situ Spacecraft Observations

Multi-spacecraft observations provide a means of assessing the time evolution of the solar-wind structure. Of particular value are pairs of observations at the same solar latitudes and radial distances, but separated in longitude by an angle \(\Delta \phi \). In this case, due to solar rotation, the spacecraft will observe the same Carrington longitude at a time difference given by \(\Delta \phi / \Omega \), where \(\Omega \) is the angular rotation speed of the Sun. Any difference between the solar-wind properties observed at the two spacecraft then equates solely to the time evolution of the solar-wind structure at the observation distance, \(r\). For long time scales (i.e. 1 hour and above, as considered in this study), this is primarily the result of changes in the solar-wind sources. Deterministic in-transit effects, such as large-scale stream interactions, will produce the same conditions at \(r\) for the same solar-wind source structure. It is only stochastic in-transit processes between the corona and spacecraft, such as turbulence (Bruno and Carbone, 2005), that can produce different conditions at \(r\) for the same solar-wind sources, but these differences are likely to be at smaller temporal/spatial scales than considered here.

The STEREO (Solar Terrestrial Relations Observatory) mission, launched in 2007 (Kaiser, 2005), consists of two spacecraft in Earth-like orbits about the Sun. They separated ahead and behind the Earth in its orbit by 22∘ per year. Thus, the STEREO spacecraft, in conjunction with near-Earth spacecraft, provide a close approximation to the ‘ideal’ dataset described above. However, due to orbits being in the ecliptic plane (which is inclined to the solar equator by an angle of 7.25∘, resulting in differences in solar latitude between spacecraft) and the longitudinal separation of the spacecraft progressing in step with the solar cycle, it is difficult to isolate the effects of time evolution, latitudinal differences and the solar cycle from this dataset (Turner et al., 2021).

Instead, we here use only the OMNI dataset of near-Earth solar-wind observations (King and Papitashvili, 2005). Earth encounters the same Carrington longitude every 27.27 days. This is the basis for so-called ‘recurrence’ (or ‘27-day persistence’) forecasts of the solar wind (e.g. Bartels, 1934; Owens et al., 2013). Between such consecutive observations, Earth also moves in solar latitude by an angle \(\Delta \theta \). Thus, a difference in solar-wind properties could be the result of time evolution and/or spatial structure in latitude. Over 27.27 days, Earth has \(\Delta \theta \leq 3.5^{\circ}\), thus we expect the \(\Delta \theta \) effect on the observed solar-wind properties to be relatively small (Owens et al., 2019; Laker et al., 2021; Turner et al., 2021).

Figure 2 shows an example of the evolution of near-Earth solar-wind structure over consecutive Carrington rotation (CR) numbers 1970 and 1971. This interval spans 23 November 2000 to 17 January 2001 and thus is close to solar maximum. The data are plotted as time through the CR, allowing the same Carrington longitudes to be directly compared. The radial magnetic field, \(B_{R}\), shows a somewhat similar four-sector structure in both CRs. Heliospheric current sheet (HCS) crossings, seen as persistent changes in \(B_{R}\) polarity, occur around days 1, 7, 12 and 20 – 22 through the CR. Looking at the magnitude of the difference in \(B_{R}\) between the two CRs, \(|\Delta B_{R}|\), the largest sustained values are around the times of these HCS crossings, suggesting some evolution of the large-scale heliospheric magnetic-field structure. The exception to this is the period of large \(|\Delta B_{R}|\) around days 0 to 6. These periods have been independently identified as ICMEs (Richardson and Cane (2010); see also https://izw1.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm). Such transient disturbances are not expected to repeat in successive Carrington rotations.

Figure 2
figure 2

Near-Earth solar-wind observations as a function of time through Carrington rotations 1970 (panels a and b) and 1971 (panels c and d). This period spans from 23 November 2000 to 17 January 2001. Solar-wind speed, \(V\) (a and c) and radial magnetic field, \(B_{R}\) (b and d) are shown, with times identified as ICMEs in black. Panels e and f show the absolute difference between the two Carrington rotations, \(|\Delta V|\) and \(|\Delta B_{R}|\), respectively.

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For the solar-wind speed, \(V\), the difference between the two CRs is more marked. Slow solar-wind is present around the four HCS crossings in both CRs, but the two fast streams in CR 1970 are absent in CR 1971. The first fast stream in CR 1970, arriving around day 3, is likely to have been the result of an ICME and thus would not be expected to repeat in CR 1971. The second fast stream arriving on day 13, however, appears to be a change in the ambient solar-wind speed structure. Thus, even when removing ICMEs from the comparison, there is a large \(|\Delta V|\) period around days 13 to 19.

Computing the average values across this particular Carrington rotation gives \({<|\Delta V|>_{CR} = 95.4}\) km s−1, reducing to 78.8 km s−1 when ICMEs are removed (i.e. removing the whole interval between the ICME first disturbance time and the ICME trailing-edge time listed in the Richardson and Cane, 2010 online catalogue), and \(<|\Delta B_{R}|>_{CR} = 3.23\) nT, reducing to 2.78 nT when ICMEs are removed.

Figure 3 shows the same analysis applied to the OMNI dataset over the period 21 November 1994 to 18 June 2020, for which there is near-complete data coverage. For solar-cycle context, the monthly sunspot number (SSN) is shown. We also construct a solar-activity index (SAI), aimed at characterising the progression of the solar cycle independent of the variation in sunspot-cycle amplitude:

$$ \mathrm{SAI} (t) = \frac{SSN_{13} (t)}{\max \left [ SSN_{13} (t - 5.5~\text{yr}: t + 5.5~\text{yr}) \right ] }, $$
(1)

where \(SSN_{13} (t)\) is the 13-month smoothed sunspot number centred on time \(t\). It is computed as a rolling average at monthly intervals. This is normalised by the maximum \(SSN_{13}\) value in an 11-year window centred on \(t\). Thus, by construction, SAI has a minimum value of 0 and reaches a maximum value of 1 in every solar cycle. Note that the 11-year window means that SAI can only currently be computed up to 2016. In order to extend the sequence, we assume that the sunspot number in the first half of Solar Cycle (SC) 25 will not exceed the maximum value of SC24. Given we are only concerned with SAI being above/below 0.5 to define high/low activity, and SAI has already dropped below 0.5 in late 2016, this assumption is highly unlikely to affect the results shown here.

Figure 3
figure 3

Time series of solar-wind properties over the period 1995 – 2020 using the OMNI dataset. (a) Monthly sunspot number (black), scaled by a factor 1/250 for plotting purposes. Also shown in red is the solar activity index (SAI) produced by scaling the 13-month smoothed sunspot number to the maximum value in an 11-year window. (b) \(<|\Delta V|>_{CR}\), the average change in solar-wind speed over a Carrington rotation. (c) \(<|\Delta B_{R}|>_{CR}\), the average change in radial magnetic field over a Carrington rotation. Blue lines show all data, red lines show values with ICMEs removed.

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Using the OMNI dataset, the average change in solar-wind speed over a Carrington rotation, \(<|\Delta V|>_{CR}\), varies from around 30 km s−1 to 150 km s−1 (see also Table 1). Whilst there is a great deal of scatter, there does appear to be a weak solar-cycle variation, quantified further below. Removing ICMEs from the data generally reduces \(<|\Delta V|>_{CR}\), as expected. ICME removal produces a greater reduction in \(<|\Delta V|>_{CR}\) at times of high solar activity than low, as there are simply more ICMEs at that time (Richardson and Cane, 2010). Thus, the effect of removing ICMEs is to reduce the overall solar-cycle variation (see also Kohutova et al., 2016).

Table 1 Mean values of \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) for both OMNI data and HelioMAS (Riley et al., 2012). Uncertainty ranges are one standard error on the mean.
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For the radial magnetic field, \(<|\Delta B_{R}|>_{CR}\) varies between approximately 2 and 5 nT. There is a more obvious solar-cycle trend. Also, note that \(<|\Delta B_{R}|>_{CR}\) is generally lower in SC24 than during the larger-amplitude SC23. Again, the effect of removing ICMEs is to reduce \(<|\Delta B_{R}|>_{CR}\) preferentially at solar maximum and hence reduce the overall solar-cycle trend.

Table 2 summarises the linear correlation coefficient (\(r_{L}\)) and the Spearman rank-ordered correlation coefficient (\(r_{S}\)) with both SSN and SAI. Using a Fisher r-to-z transformation (Press et al., 1989) with N = 342, all reported correlations are found to be significant (strictly speaking, the probability is below 1% that the reported correlation is consistent with zero for the given sample size).

Table 2 Correlation coefficients of \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) with SSN and SAI. Linear correlation coefficients (\(r_{L}\)) are shown with Spearman rank-ordered correlation coefficient (\(r_{S}\)) in parentheses.
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Figure 4 summarises the distributions of \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) for different divisions of the data. Box and whisker plots are used, where the white line shows the median, the coloured area spans with interquartile range (IQR) and the thin line spans the maximum to minimum values excluding outliers (i.e. 1.5 IQR above and below the third and first quartiles, respectively). The notch about the median shows the 95% confidence interval on the median.

Figure 4
figure 4

Box and whisker plots summarising the distributions of \(<|\Delta V|>_{CR}\) (panels a and b) and \(<|\Delta B_{R}|>_{CR}\) (c and d) obtained from consecutive 27-day intervals of OMNI observations. The white line shows the median, the coloured area spans with interquartile range (IQR), and the thin line spans the maximum to minimum values excluding outliers (i.e. 1.5 IQR above and below the third and first quartiles, respectively). The notch about the median shows the 95% confidence interval on the median. The left-hand panels (a and c) show all OMNI data for the period 21 November 1994 to 18 June 2020, the right-hand panels (b and d) show the same period with ICMEs removed. Black, blue, and red show all, low, and high solar-activity levels, respectively.

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A number of features are worth noting. First, \(<|\Delta B_{R}|>_{CR}\) exhibits a stronger correlation with both measures of the solar cycle than \(<|\Delta V|>_{CR}\). Secondly, the correlation of both \(<|\Delta B_{R}|>_{CR}\) and \(<|\Delta V|>_{CR}\) with SSN is stronger than the equivalent with SAI. From this we infer that the amplitude – as well as phase – of the solar cycle affects the observed rate of change of the solar-wind structure.

Taking SAI = 0.5 as the threshold between low and high solar activity, both \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) show significantly higher values during high SAI. However, the magnitude of this solar-cycle variation is small (of the order of 10%); see also Table 1.

For both \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\), removing ICMEs further reduces the magnitude of the solar-cycle variation. Thus, much of the observed relation between solar-wind variability and solar cycle is the result of increased occurrence of transient ICMEs at solar maximum. Whilst the differences between high and low activity remain statistically significant, for the ICME-removed series, \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) increase by only 8% between low and high solar activity.

Using the OMNI dataset we cannot completely remove the effect of latitudinal offset, \(\Delta \theta \) (see also Turner et al., 2021) or transients resulting from either unidentified ICMEs and/or turbulent and mesoscale structures (Viall, DeForest, and Kepko, 2021; Verscharen, Klein, and Maruca, 2019). Thus, it is useful to also examine simulation results that attempt to capture only the ambient solar wind. This also allows a more global picture of changing solar-wind structure.

3 Solar-Wind Simulations

Global coronal models, constrained by the photospheric magnetic observations, are typically used to approximate the steady-state corona. These coronal solutions can then be used to drive heliospheric models out to Earth orbit. Potential-field source surface (PFSS) models are intrinsically steady state (Altschuler and Newkirk, 1969; Schatten, Wilcox, and Ness, 1969), whereas models solving the time-dependent MHD equations are relaxed until a steady-state equilibrium is achieved. Indeed, triggering a CME-like eruption is often achieved by ad hoc modifications of the boundary conditions in order to energise the system (e.g. Torok et al., 2018).

This section uses archived output from the combined Magnetohydrodynamics Algorithm outside a Sphere (MAS: Linker et al., 1999) coronal and HelioMAS (Riley et al., 2012) solar-wind MHD models, using a range of photospheric observatories. All data used are available from https://www.predsci.com/mhdweb/home.php.

Example solutions at 1 AU for CRs 2100 and 2101 using magnetograms from the Helioseismic and Magnetic Imager (HMI: Scherrer et al., 2012) are shown in Figure 5. This period spans from 9 August 2010 to 3 October 2010 and thus is close to solar-minimum conditions. Both the \(V\) and \(B_{R}\) structures for the two CRs show a reasonable agreement at the global scale. In both CRs, fast wind is confined to the polar regions, with a broad band of slow wind spanning the equator, but slightly skewed towards the southern hemisphere. The HCS, seen as the white line separating opposite \(B_{R}\) polarities, shows two large excursions into the southern hemisphere around Carrington longitudes of 180∘ and 330∘ in both CRs.

Figure 5
figure 5

Latitude–longitude maps at a constant radius of 1 AU for heliospheric MHD solutions constrained by HMI magnetograms. Carrington rotations 2100 (panels a and b) and 2101 (panels b and c) are shown. Solar-wind speed, \(V\), is shown in the left column, with radial magnetic field, \(B_{R}\), on the right. The absolute difference between the two CRs is shown in panels e and f. Dashed horizontal lines span a latitude of \(+/- 7.25^{\circ}\) about the solar equator.

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Taking the difference between consecutive steady-state coronal solutions has previously been used as a method to identify regions of coronal evolution (Luhmann et al., 1998, 1999). This evolution has been associated with CMEs, suggesting coronal eruptions may be a way in which the corona transitions between quasi-steady-state conditions (see also Low, 2001; Owens and Crooker, 2006; Owens et al., 2007). Here, we look at the change in the steady-state 1-AU solar-wind properties. Despite the global-scale agreement between CRs 2100 and 2101, Figures 5e and f show that there are nevertheless regions of large difference, in both \(V\) and \(B_{R}\). These are most pronounced in the mid-latitudes. Around the ecliptic-plane latitudes – the band between −7.25 and 7.25∘ from the equator – \(|\Delta V|\) is relatively suppressed, as both CRs largely show fairly uniform slow wind at those latitudes. \(|\Delta B_{R}|\), on the other hand, shows similar values at mid-latitudes and the ecliptic.

Averaging over all latitudes and longitudes for this Carrington rotation, we find \(<|\Delta V|>_{CR} = 73.0\) km s−1 and \(<|\Delta B_{R}|>_{CR} = 0.509\) nT. When restricted to ecliptic longitudes, \(<|\Delta V|>_{CR} \) drops by around a quarter to 58.2 km s−1, whereas \(<|\Delta B_{R}|>_{CR}\) increases to 0.888 nT.

Figures 6 and 7 show the same analysis applied to all available HelioMAS solutions, spanning the years 1974 to 2020. Magnetograms from a range of observatories are used, but \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) are only computed when HelioMAS solutions are available that are based on magnetograms from the same observatory for two consecutive CRs. For the purpose of computing average values over the whole time period, a composite series is also produced. This uses all observatories, and when multiple observatories are available for the same interval, an average value of \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) is computed. These numbers are reported in Tables 1 and 2 and shown in Figure 8.

Figure 6
figure 6

Time series of \(<|\Delta V|>_{CR}\) from consecutive CRs obtained from HelioMAS MHD simulations at 1 AU. (a) Scaled monthly sunspot number and the solar-activity index, shown for solar-cycle context. (b) \(<|\Delta V|>_{CR}\) averaged over all latitudes. The colour coding for different observatories is given in the legend. See https://www.predsci.com/mhdweb/home.php for more details. (c) \(<|\Delta V|>_{CR}\) averaged over the \(+/- 7.25^{\circ}\) about the solar equator.

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Figure 7
figure 7

Same as Figure 6, but for \(<|\Delta B_{R}|>_{CR}\), i.e. time series of \(<|\Delta B_{R}|>_{CR}\) from consecutive CRs obtained from HelioMAS MHD simulations at 1 AU. (a) Scaled monthly sunspot number and the solar-activity index, for solar-cycle context. (b) \(<|\Delta B_{R}|>_{CR}\) averaged over all latitudes. The colour coding for different observatories is given in the legend. See https://www.predsci.com/mhdweb/home.php for more detail. (c) \(<|\Delta B_{R}|>_{CR}\) averaged over the \(+/- 7.25^{\circ}\) about the solar equator.

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Figure 8
figure 8

Box and whisker plots summarising the distributions of \(<|\Delta V|>_{CR}\) (panels a and b) and \(<|\Delta B_{R}|>_{CR}\) (c and d) obtained from consecutive solutions of HelioMAS at 1 AU. The white line shows the median, the coloured area spans with interquartile range (IQR), and the thin line spans the maximum to minimum values excluding outliers (i.e. 1.5 IQR above and below the third and first quartiles, respectively). The left-hand panels (a and c) show values averaged over all latitudes, the right-hand panels (b and d) show values at ecliptic latitudes (i.e. \(+/- 7.25^{\circ}\) about the solar equator). Black, blue, and red show all, low, and high solar-activity levels, respectively.)

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For solar-wind speed, the global (i.e. over all latitudes) average \(<|\Delta V|>_{CR}\) ranges from approximately 20 to 150 km s−1. There is a clear solar-cycle variation. Although there is a large data gap during SC23, there is not an obvious cycle-to-cycle variation in the peak values of global \(<|\Delta V|>_{CR}\). For example, the low-amplitude SC24 shows similar \(<|\Delta V|>_{CR}\) values to the large-amplitude SC19 (around 1980). This is supported by the correlation coefficients summarised in Table 2: the correlation of global \(<|\Delta V|>_{CR}\) is stronger with SAI – which removes the effect of changing solar-cycle amplitude – than it is with SSN. This trend is opposite to that observed with the OMNI dataset, which suggests there may be an increased occurrence of unidentified transients in the OMNI dataset during strong solar cycles.

Limiting analysis to ecliptic latitudes, \(+/- 7.25^{\circ}\) about the solar equator, gives a different picture; there is little solar-cycle variation in \(<|\Delta V|>_{CR}\). The average value of \(<|\Delta V|>_{CR}\) at ecliptic latitudes is in approximate agreement with that obtained from the OMNI dataset.

Global \(<|\Delta B_{R}|>_{CR}\) shows a strong correlation with the solar cycle. As 1-AU \(|B_{R}|\) is expected to be higher in higher-amplitude solar cycles (Lockwood, Stamper, and Wild, 1999; Solanki, Schüssler, and Fligge, 2000; Owens and Lockwood, 2012), it is perhaps unsurprising that the correlation of \(<|\Delta B_{R}|>_{CR}\) is slightly higher with SSN than SAI. Confining this analysis to ecliptic latitudes, the correlation of \(<|\Delta B_{R}|>_{CR}\) with both SSN and SAI is reduced, but still significant and of large amplitude. Note that the magnitude of \(<|\Delta B_{R}|>_{CR}\) is lower than that obtained from the OMNI dataset, as magnetogram-based estimates of the heliospheric magnetic field are known to underestimate \(|B_{R}|\) by around a factor two (Wallace et al., 2019; Linker et al., 2017, 2021)

Taking a threshold of SAI = 0.5, global \(<|\Delta V|>_{CR}\) varies by a factor 2.39 between periods of low and high solar activity (i.e. a 139% increase during high activity). For \(<|\Delta B_{R}|>_{CR}\), we find a factor 2.12 (i.e. a 112% increase during high activity). Restricting to ecliptic latitudes reduces these solar-cycle variations to 10% for \(<|\Delta V|>_{CR}\) and 38% for \(<|\Delta B_{R}|>_{CR}\). While still larger than the values found by analysis of the OMNI data, particularly for \(<|\Delta B_{R}|>_{CR}\), these values are in closer agreement with the weak solar-cycle trends in in situ observations.

4 The Role of Latitude

In order to better understand the difference between global and ecliptic estimates of \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\), it is useful to further investigate the latitudinal structure of both the solar-wind properties and their rate of change.

Figure 9 shows how \(V\) varies with latitude and time. Panel b shows longitudinal averages of \(V\) over a CR (see also Figure 4a of Owens, Lockwood, and Riley, 2017). This is in close agreement with the structure observed both in situ (McComas et al., 2003) and via radio scintillation (Manoharan, 2012). At solar minimum, there is slow wind around the solar equator and uniform fast wind at the mid- and high latitudes. At solar maximum, slow wind extends to all latitudes.

Figure 9
figure 9

The latitudinal structure of the 1-AU solar-wind speed, \(V\), from HelioMAS as a function of time. (a) Sunspot number and solar-activity index, for solar-cycle context. (b) The longitudinally averaged \(|V|\) for each CR, \(<|V|>_{CR}\), as a function of latitude and time. Panels d and e show the standard deviation of \(V\) within a CR, \(\sigma _{V}\), as a function of latitude and time. This is a measure of the longitudinal structure in \(V\) for a given CR. Panels c and e show \(<|V|>_{CR}\) and \(\sigma _{V}\) as a function of latitude and averaged over all times (black), low solar activity (blue), and high solar activity (red).

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Figure 9e shows \(\sigma _{V}\), the standard deviation in \(V\) over a Carrington rotation. This is a measure of the degree of longitudinal structure in \(V\). At times and latitudes with uniform fast wind, \(\sigma _{V} = 0\). However, \(\sigma _{V}\) is also relatively low at latitudes and times where slow wind completely dominates, such as near the equator during solar maximum. The highest values of \(\sigma _{V}\) are produced at latitudes that contain both fast and slow wind, right at the transition between the two solar-wind regimes.

Figure 10 shows the same analysis for \(B_{R}\). Panels b and c show that \(<|B_{R}|>_{CR}\) does not show a large variation in either latitude or time. Conversely, \(\sigma _{B}\), shown in panels d and e, shows a somewhat similar variation to \(\sigma _{V}\); it is effectively zero at high latitudes during low solar activity, with large values extending to higher latitudes at high solar activity. The main difference from \(\sigma _{V}\), however, is that \(\sigma _{B}\) remains high at the equator throughout the solar cycle. This is because large \(\sigma _{B}\) is effectively a marker for latitudes and times where \(B_{R}\) changes polarity. It will thus be maximised at the equator as the HCS is, on average, centred at the equator (see, however, Mursula and Hiltula, 2003), with high-latitude excursions becoming more common at solar maximum (e.g. Smith, 1990). Thus, mixed magnetic polarities (and high \(\sigma _{B}\)) are always present at the equator (see also Figure 13 of Owens and Forsyth, 2013).

Figure 10
figure 10

The latitudinal structure of the 1-AU radial magnetic field, \(B_{R}\), from HelioMAS as a function of time. (a) Sunspot number and solar-activity index, for solar-cycle context. (b) The longitudinally averaged \(|B_{R}|\) for each CR, \(<|B_{R}|>_{CR}\), as a function of latitude and time. Panels d and e show the standard deviation of \(B_{R}\) within a CR, \(\sigma _{B}\), as a function of latitude and time. This is a measure of the longitudinal structure in \(B_{R}\) for a given CR. Panels c and e show \(<|B_{R}|>_{CR}\) and \(\sigma _{B}\) as a function of latitude and averaged over all times (black), low solar activity (blue), and high solar activity (red).

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With this global context in mind, we now consider how the rate of change of the solar wind varies as a function of latitude. Figure 11 shows \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) with latitude and time. Figure 11b shows that the overall envelope of (longitudinally averaged) \(<|\Delta V|>_{CR}\) closely follows that of \(<|V|>_{CR}\). High latitudes around solar minimum contain pure fast wind. As fast wind in HelioMAS simulations has uniform characteristics, \(<|\Delta V|>_{CR} = 0\) km s−1 at these times and locations. With increasing solar activity, slow wind extends to higher latitudes. The variability of the slow wind means \(<|\Delta V|>_{CR} > 0\) km s−1 at any latitude that contains some slow wind. The very highest values of \(<|\Delta V|>_{CR}\) occur on the boundary of fast and slow winds, closely following \(\sigma _{V}\). This is due to \(V\) at a given longitude changing between one regime and the other between consecutive CRs, and generating high \(|\Delta V|\) as the precise location and timing of that transition evolves.

Figure 11
figure 11

The effect of latitude on \(<|\Delta V|>_{CR}\) (b and c) and \(<|\Delta B_{R}|>_{CR}\) (d and e) from HelioMAS MHD simulations at 1 AU. (a) Time series of scaled SSN and SAI over the period 1975 to 2020. (b) \(<|\Delta V|>_{CR}\) as a function of latitude and time. Panel d shows the same for \(<|\Delta B_{R}|>_{CR}\). Panels c and e show \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\), respectively, as a function of latitude and averaged over all times (black), low solar activity (blue), and high solar activity (red).

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Splitting the data into high- and low-activity periods using a threshold of SAI = 0.5 shows this behaviour more clearly. In Figure 11c, \(<|\Delta V|>_{CR}\) during high activity is approximately constant with latitude at around 100 km s−1, with a slight decrease at the equator. This decrease is due to a near-complete absence of fast wind at these times/locations, reducing the possible variation (as seen by the decrease in \(V\) and \(\sigma _{V}\) at the equator during solar maximum). At low activity, however, a strong latitudinal variation in \(<|\Delta V|>_{CR}\) is present. \(<|\Delta V|>_{CR}\) reaches around 100 km s−1 around the equator, similar to the high-activity value. Thus, the correlation of \(<|\Delta V|>_{CR}\) with the solar cycle is low at the equator and high at the poles, as reported in Table 2.

\(<|\Delta B_{R}|>_{CR}\) peaks at the equator throughout the solar cycle, but the maximum amplitude is higher at times of high solar activity than low. The amplitude variation is largely due to \(<|B_{R}|>\) varying with solar activity. Thus, the correlation of \(<|\Delta B_{R}|>_{CR}\) with SSN is high (\(> 0.4\)) at all latitudes.

5 Discussion

In this study we have quantified the rate of change of the ambient solar-wind structure over consecutive Carrington rotations (i.e. 27-day periods). This is important for both understanding the degree to which the steady-state approximation is valid, and is required in order to best exploit the available in situ solar-wind observations for data assimilation.

The first approach was to use solar-wind speed, \(V\), and radial magnetic-field component, \(B_{R}\), observed in near-Earth space using the OMNI dataset. By looking at the differences in consecutive Carrington rotations (CRs), we computed the changes in solar-wind properties at the same solar longitude, \(|\Delta V|\) and \(|\Delta B_{R}|\). A weak solar-cycle trend was found in the Carrington rotation averages of both these properties, \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\). This could be partly attributed to the increased occurrence of interplanetary coronal mass ejections (ICMEs) at high solar activity. This is expected, as transient disturbances – by definition – do not repeat in each CR. After removing ICMEs there was only an 8% difference in high and low solar activity values of \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) for the ambient solar wind. This result appears to contradict the idea that coronal (and hence solar-wind) structure varies much more at solar maximum than minimum.

To obtain more of a global picture of solar-wind structure, the second half of the study used coronal and heliospheric MHD simulations constrained using the observed photospheric magnetic field. Such simulations are steady state and approximate the ambient solar-wind conditions only (CMEs are typically added as an additional, time-varying perturbation, e.g. Odstrcil, Riley, and Zhao, 2004). Globally, both \(<|\Delta V|>_{CR}\) and \(<|\Delta B_{R}|>_{CR}\) show a very strong correlation with solar activity, both in terms of the statistical significance of the relation and the magnitude of the effect (around a factor two). Restricting the analysis to ecliptic latitudes drastically reduces this trend, to 10% for \(<|\Delta V|>_{CR}\) and to 38% for \(<|\Delta B_{R}|>_{CR}\). Whilst these values are still higher than those obtained from the in situ observations, they are in much closer agreement.

Thus globally, the rate of change of solar-wind speed structure does correlate with the solar cycle, as expected. Specifically, \(<|\Delta V|>_{CR}\) correlates more closely with the phase of the solar cycle than with sunspot number, which is determined by both phase and solar-cycle amplitude. Given solar-wind speed is partially ordered by the position of the heliospheric current sheet (HCS), this agrees with the findings that the average HCS position is a function of solar-cycle phase, but not solar-cycle amplitude (Alanko-Huotari et al., 2007; Owens and Lockwood, 2012). Note, however, that this relation may break down in very weak cycles (Owens, Lockwood, and Riley, 2017), possibly due to weakening of the photospheric magnetic field and the relation between coronal magnetic-field topology and solar-wind speed (Wang and Sheeley, 1990; Riley, Linker, and Arge, 2015).

Conversely, the global variability in the radial field is more closely correlated with the sunspot number than solar-cycle phase alone. This is likely the result of heliospheric radial magnetic-field strength varying closely with sunspot-cycle magnitude, as seen through solar open solar flux reconstructions (Lockwood, Stamper, and Wild, 1999; Solanki, Schüssler, and Fligge, 2000; Krivova, Balmaceda, and Solanki, 2007; Owens and Lockwood, 2012).

\(<|\Delta V|>_{CR}\) generally peaks at mid-latitudes, where slow and fast winds often coexist, giving the possibility of the largest \(|\Delta V|\) values. At the equator, fast wind is more rarely present, reducing the possibility of large \(|\Delta V|\). Conversely, \(<|\Delta B_{R}|>_{CR}\) peaks at the equator, owing to the HCS – and hence mixed \(B_{R}\) polarity – being centred at the equator.

For the purposes of solar-wind data assimilation, it is necessary to relate the reliability of observations to their age. The results presented here suggest that, at least for in-ecliptic in situ observations, it is not necessary to assume that the rate at which observations become less reliable is a function of the solar cycle. Instead, we suggest simply using an ‘age’ uncertainty in solar-wind speed of around 70 km s−1 per CR (i.e. 3 km s−1 per day) since the observation was made, and 3 nT per CR (i.e. 0.1 nT per day) for the radial magnetic field. These would be added to other contributions to the representivity error. For observations from higher latitudes, such as from interplanetary scintillation or heliospheric imager observations, this age uncertainty is more strongly solar-cycle dependent.