Abrupt Shrinking of Solar Corona in the Late 1990s

Magnetic field of the visible solar surface, the photosphere, determines the structure of the solar corona, solar wind, and the heliospheric magnetic field (HMF; Cahill 1965) and, through them, controls the disturbances affecting the near-Earth space, for example, geomagnetic storms and aurorae. The photospheric magnetic field intensifies when new active regions like sunspots appear on the solar surface (Wang et al. 1989; Smith et al. 2003). The number of sunspots varies in cycles of 10–11 yr, but the height of these sunspot cycles also varies from cycle to cycle, indicating longer-term variability in solar activity. The height of sunspot cycles increased systematically during the first half of the twentieth century, leading to a maximum during cycle 19 in the late 1950s, now called the Grand Modern Maximum (GMM). Solar activity has declined since then, first rather slowly during cycles 20–22 but then, during the current cycle 24, the Sun's activity slowed to a lower level that was typical one hundred years ago, before the start of the GMM.

The magnetic field of sunspots (Hale et al. 1919) and the activity of the solar chromosphere (Ellerman 1919) were observed already since the rising phase of the GMM. However, the information on solar corona during those early decades is mainly based on proxy data like geomagnetic activity (Mursula et al. 2015, 2017) and momentary snapshot observations of the corona during solar eclipses (Tlatov 2010). Direct satellite observations of the solar wind started in the early 1960s, allowing for a greatly improved understanding of the structure of the HMFs. Understanding solar magnetic fields was further aided by continuous magnetograph observations of the photospheric magnetic field, which started soon after the start of the satellite era. Accordingly, almost the entire declining phase of GMM is covered by versatile magnetic observations, which now allow us to study the effects of this long-term decline of solar activity on the corona.

We know roughly how the structure of the coronal magnetic field is formed by the photospheric magnetic field (Petrie 2013; Wiegelmann et al. 2017; Mikić et al. 2018; Cheung et al. 2019), but the relation between the photosphere and the corona is complicated and time-variable. Moreover, because we still cannot directly measure the coronal magnetic field, most information on the coronal field is based on models using photospheric observations as their input. The most commonly used model of the coronal magnetic field is the potential field source surface (PFSS) model (Altschuler & Newkirk 1969; Schatten et al. 1969; Hoeksema et al. 1983), which assumes that there are no significant electric currents between the photosphere and the coronal source surface, and that the coronal magnetic field becomes radial at the source surface distance of about 1.5–3.5 solar radii (R_s). With these assumptions the coronal magnetic field between the photosphere and the source surface can be solved using the observed photospheric magnetic field as a boundary condition.

Several studies compare the coronal magnetic field derived from the PFSS model with heliospheric observations. Riley et al. (2006) concluded that magnetohydrodynamic (MHD) models require a lot of calculation power (and are required for modeling time-dependent structures), but yield rather little advance to the PFSS model, despite its simplicity. They noted that choosing the data set for the photospheric field may have a larger effect than choosing between the PFSS or the MHD model. Lee et al. (2011) compared coronal holes and open coronal flux derived from the PFSS model to coronal holes from extreme-ultraviolet (EUV) images and heliospheric observations of the open HMF flux. They derived the value of the source surface radius to be roughly the same, about 1.9 and 1.8 R_s, during the minima after solar cycles 22 and 23, respectively. Arden et al. (2014) concluded that a constant source surface of 2.5 R_s can reproduce the observed HMF flux during cycle 23. However, we have not been able to reproduce these results. Luhmann et al. (2013) used Global Oscillation Netwrok Group (GONG) data and both the PFSS model and the magnetohydrodynamics around a sphere (MAS) model to study the coronal geometry and solar wind sources during solar cycle 23, finding a fairly good agreement between the two models. Recently, (Koskela et al. 2017) showed that the PFSS model reliably produces the HMF sector structure. Moreover, adding horizontal currents (as in the current sheet source surface (CSSS) model; Koskela et al. 2019) or using an MHD model (Li & Feng 2018) does not improve the polarity match between the corona and the heliosphere. All these studies indicate that, despite its simplified assumptions (lack of currents and spherical source surface), the PFSS model can well produce the large-scale coronal magnetic field structure, even at different solar activity conditions.

The absolute value of the radial component of the HMF, also called the (unsigned) HMF flux density ϕ_HMF is a measure of magnetic energy emitted by the corona with the solar wind into the heliosphere. Because the total magnetic flux is conserved, the HMF flux through a sphere around the Sun is invariant of the sphere radius, and the initial HMF flux density at the site of HMF emission, the coronal source surface, decreases with radial distance inversely proportional to the area of a sphere (1/r²). Moreover, as HMF observations by the Ulysses probe over a large range of heliographic latitudes have shown that the HMF flux density is independent of latitude (Smith & Balogh 1995; Owens et al. 2008), the total HMF flux can be determined by a flux density measurement at any single point in the heliosphere.

A major challenge of studying the long-term evolution of the solar magnetic fields is imposed by the fact that the different magnetograph instruments yield quite different photospheric flux densities depending, e.g., on spectral and spatial resolution, data processing method, and the treatment of the unobserved solar poles (Riley et al. 2014). We have recently developed a new method for scaling the magnetograms from different instruments to each other in terms of harmonic expansion (Virtanen & Mursula 2017, 2019). This method can be used to scale any pair of photospheric field observations with a reasonable overlapping period.

In this Letter we study the long-term evolution of the coronal field using the PFSS model and Wilcox Solar Observatory (WSO) and the Mount Wilson Observatory (MWO) separately to the more recent high-resolution observations by the Synoptic Optical Long-term Investigations of the Sun/Vector Spectro Magnetograph (SOLIS/VSM). We compare the total flux of the coronal field to the HMF measured at Earth's orbit. The Letter is structured as follows. In Section 2 we present the data sets and methods we use. In Section 3 we discuss the long-term evolution of the coronal field, and in Section 4 we summarize our results.

We use hourly HMF and SW data observed around the Earth's orbit by several satellites since the 1960s. These data are scaled to 1 au, and collected to the NASA/NSSDC OMNI-2 database (King & Papitashvili 2005). These data provide the longest single-point measurement in the heliosphere with variable, but mostly sufficient data coverage. In order to study long-term variations, we use here HMF flux density averages over one solar rotation (Carrington period of 27.2753 days). When calculating the rotational averages in this article we require a 20% data coverage of hourly values (minimum of 131 values out of 654) for each rotation. If the coverage is smaller than this, the corresponding rotation is neglected.

We use WSO synoptic maps of the photospheric magnetic field since 1976. WSO magnetograph is a low-resolution device (3' aperture size), and suffers from saturation (Svalgaard et al. 1978). This affects the intensity of the measured field, but the erroneous absolute level has largely been corrected by the new scaling method (Virtanen & Mursula 2017). The same instrumentation has been operating at WSO since the start of observations, which makes the WSO database homogeneous and, therefore, very useful for long-term studies. We neglect rotations 1905–1944 (1996 January–1998 December) and 1973–1978 (2001 February–2001 June) when WSO data is known to be erroneous (Virtanen & Mursula 2016).

As a second data set of the photospheric magnetic field we use the calibrated synoptic maps since 1967 from the MWO. Note that the data of the period of 1967–1974 became publicly available only in 2018, and have not been used for long-term studies so far. There were instrumental updates at MWO in 1974, 1982, 1994, and 1996, and MWO observations were terminated in 2013 January. We use only full synoptic maps (with no data gaps) in their original published format without any modifications. (For more detailed description of magnetograph data, see Virtanen & Mursula 2016, 2017 and references therein.)

We scale the WSO and MWO data sets to the absolute level of the SOLIS/VSM instrument, which measured the photospheric magnetic field in 2003–2017. In the harmonic scaling method (Virtanen & Mursula 2017), a linear regression is derived between the corresponding harmonic coefficients of ${g}_{n}^{m}$ and ${h}_{n}^{m}$ (see Section 2.2) for any two data sets using rotations included in the overlapping time interval. For each pair of data sets, we calculate the scaling matrices, for example, G^m_n(MWO → SOLIS) and H^m_n(MWO → SOLIS), which are used to multiply the harmonic coefficients of the data set to be scaled (MWO) to the level of the other data set (SOLIS). We note that the G^m_n and H^m_n are constant in time. We hereafter denote the WSO (MWO) data scaled to SOLIS/VSM as WSO_VSM (MWO_VSM)

2.1. Heliospheric Flux Density

Ulysses probe observations (Smith & Balogh 1995) have shown that $| {B}_{r}|$ is independent of latitude. Using this, together with the radial decay of B_r ∼ r⁻², we end up with the following form for the total unsigned flux:

$$\begin{eqnarray}&&\phi =4\pi \langle {r}^{2}| {B}_{r}| {\rangle }_{27d},\end{eqnarray} \tag{ 1 }$$

where the average is taken over one solar (Carrington) rotation of 27.2753 days.

There are different methods to derive the magnetic flux density in the heliosphere. The challenge is that the distribution of B_r includes both positive and negative values. A simple average of $| {B}_{r}|$ overestimates the unsigned flux in case of high-resolution data, mainly because of noise, and underestimates it in case of very low-resolution data, where opposite HMF sectors (polarities) are effectively averaged (Smith 2011).

Erdös & Balogh (2012) suggested that all deviations from the Parker spiral field are fluctuations that should be removed before calculating the rotational averages of the radial field. In order to remove these fluctuations, the rotational averages of $| {B}_{r}|$ are calculated from the component aligned along the Parker spiral field.

Parker's model for the HMF is

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{P}={B}_{\mathrm{ss}}{\left(\displaystyle \frac{{r}_{\mathrm{ss}}}{r}\right)}^{2}\left(\hat{{\boldsymbol{r}}}+\displaystyle \frac{{v}_{\phi }-r{\rm{\Omega }}}{{v}_{r}}\cos \lambda \hat{\phi }\right),\end{eqnarray} \tag{ 2 }$$

where B_ss is the source surface flux density, r_ss is the source surface radius, v_r and v_ϕ are the radial and azimuthal velocity components of velocity, r is the heliocentric distance, λ is the heliographic latitude, and Ω the solar angular velocity. The HMF component along Parker's spiral is then

$$\begin{eqnarray}&&{{\boldsymbol{B}}}^{* }=\displaystyle \frac{{\boldsymbol{B}}\cdot {{\boldsymbol{B}}}_{P}}{| {{\boldsymbol{B}}}_{P}{| }^{2}}{{\boldsymbol{B}}}_{P}\end{eqnarray} \tag{ 3 }$$

and the rotational averages of the unsigned HMF flux ${\phi }_{\mathrm{au}}^{\mathrm{EB}}$ are obtained as follows:

$$\begin{eqnarray}&&{\phi }_{\mathrm{au}}^{\mathrm{EB}}=\langle | {{\boldsymbol{B}}}^{* }\cdot \hat{{\boldsymbol{r}}}| {\rangle }_{27d}.\end{eqnarray} \tag{ 4 }$$

(Note that six-hour averages of ${\boldsymbol{B}}$ and ${{\boldsymbol{B}}}^{P}$ are used in Equation (3) in order to reduce fluctuations, as suggested by Erdös & Balogh 2012).

Lockwood et al. (2009) developed a sophisticated method of removing kinematic effects, which relates to variable solar wind velocity and the tangential magnetic field component. However, they concluded that a more simple method of first deriving 24 hr averages of signed B_r and then averaging over the solar rotation time gives close to the same result as the sophisticated method. We use here this simple method in order to calculate another version of the HMF flux density at 1 au as follows:

$$\begin{eqnarray}&&{\phi }_{\mathrm{au}}^{L}=\langle | \langle {B}_{r}{\rangle }_{24h}| {\rangle }_{27d}.\end{eqnarray} \tag{ 5 }$$

Figure 1 shows ϕ_au according to the methods by Lockwood et al. (2009) and by Erdös & Balogh (2012). The Lockwood method yields a systematically larger ${\phi }_{\mathrm{au}}$ but the difference between the two methods is very small (about 0.12 nT).

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2.2. PFSS Model

The PFSS model assumes that, at a certain distance called the source surface (r_ss), the radially out-flowing plasma takes over the magnetic field and the field becomes radial. Thereby, the outer boundary condition of the PFSS model requires that the field is radial at r_ss. The PFSS solution for the radial component of the coronal magnetic field between the photosphere and the source surface is

$$\begin{eqnarray}\begin{array}{rcl}{B}_{r}(r,\theta ,\phi ) & = & \sum _{n=1}^{9}\sum _{m=0}^{n}{P}_{n}^{m}(\cos \theta )\\ & & \times ({g}_{n}^{m}\cos m\phi +{h}_{n}^{m}\sin m\phi )C(r,n),\end{array}\end{eqnarray} \tag{ 6 }$$

where the radial functions C(r, n) are

$$\begin{eqnarray}&&C(r,n)={\left(\displaystyle \frac{{R}_{s}}{r}\right)}^{n+2}\left[\displaystyle \frac{n+1+n{\left(\tfrac{r}{{r}_{\mathrm{ss}}}\right)}^{2n+1}}{n+1+n{\left(\tfrac{{R}_{s}}{{r}_{\mathrm{ss}}}\right)}^{2n+1}}\right],\end{eqnarray} \tag{ 7 }$$

and ${P}_{n}^{m}(\cos \theta )$ are the associated Legendre functions, gⁿ_m and h^m_n are the harmonic coefficients of the spherical harmonic expansion, R_s is the solar radius, r is the radial distance, and θ is the co-latitude (polar angle). (The nonphysical magnetic monopole term n = 0 is neglected.) For a more detailed description of the PFSS model and the method of deriving harmonic coefficients g^m_n and h^m_n, see Virtanen & Mursula (2016).

Here we use the long-term photospheric data series WSO_VSM and MWO_VSM as input to the PFSS model in order to study the long-term evolution of the coronal magnetic field. The larger the source surface radius of the PFSS model is, the more of the coronal field lines close within the corona and the smaller the HMF flux density will remain. This is seen in Figures 2(a) and (b) where we have plotted the mean coronal magnetic flux density (${\phi }_{\mathrm{ss}}$) scaled from the coronal source surface to 1 au for three coronal source surface distances (${r}_{\mathrm{ss}}=1.5\,{R}_{s},2.5\,{R}_{s},3.5\,{R}_{s}$) for WSO since 1976 and for MWO in 1967–2013. (Source surface fluxes were scaled to the distance of 1 au using the $1/{r}^{2}$ radial decrease of the flux density). The overall mean values of the flux densities ϕ_ss for the three r_ss values of 3.7, 1.9, and 1.3 nT for WSO_VSM, and 3.8, 2.0, and 1.4 nT for MWO_VSM. (The small difference is due to the somewhat different time intervals). Figures 2(a) and (b) also include the ${\phi }_{\mathrm{au}}^{\mathrm{EB}}$ reproduced from Figure 1. The mean of ${\phi }_{\mathrm{au}}^{\mathrm{EB}}$ over the whole time interval is 2.7 nT, showing that the overall best-fitting value of r_ss is between 1.5 and 2.5 R_s.

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Figures 1, 2(a) and (b) show that the maxima of ϕ_au occur in the early declining phase of solar cycles 21–24 (in 1982, 1991, 2003, and 2014/2015) and the minima of ϕ_au around sunspot minima. (ϕ_au shows only a weak peak in 1974 during solar cycle 20). The timings of the ϕ_ss maxima agree fairly well with ϕ_au maxima for all r_ss values, but the timings of ϕ_ss minima are quite different for different r_ss values (Virtanen & Mursula 2019). For r_ss = 1.5 R_s MWO_VSM shows minima around sunspot minima in 1986, 1996, and 2009 and WSO_VSM in 1987 and 2009, in a good agreement with ϕ_au. However, for r_ss = 2.5 R_s and r_ss = 3.5 R_s neither MWO_VSM nor WSO_VSM show minima at the same time as ϕ_au, i.e., at sunspot minima, but only much later, in the ascending phase or even at the maximum of the next cycle.

The maximum-time (minimum-time) level of ϕ_au has systematically decreased from cycle 22 maximum to cycle 24 maximum (from mid-1980s to mid-2000s, respectively). This long-term decrease of the observed HMF flux density in recent decades is well documented (Smith & Balogh 2008; Lockwood 2013). The same systematic decline is seen in ϕ_ss in Figures 2(a) and (b) for all values of r_ss from cycle 22 until recently. However, the relative reduction is clearly larger in ϕ_ss than in ϕ_au during the same time interval. For r_ss = 1.5 R_s the corresponding flux density ϕ_1.5 agrees very well with ϕ_au during the last 20 yr but greatly exceeds that in the earlier decades. The closest match in the 1980s is found between ϕ_2.5 and ϕ_au but ϕ_2.5 remains significantly below ϕ_au during the last 20 yr. These systematic differences between ϕ_ss and ϕ_au reflect the fact that the long-term evolution of the modeled coronal field ϕ_ss does not fit with the observed flux density ϕ_au using any constant value of the source surface radius. The same conclusion was made by Wang et al. (2000) ,who used a constant r_ss for the early decades, but noted that cycle 23 cannot be reproduced with the same value.

In order to solve this problem we now calculate ϕ_ss so that r_ss does not have to be constant but can vary in time. For each rotation, starting from r_ss = 1.05 R_s (for which ϕ_ss exceeds ϕ_au for all rotations) we increase r_ss in steps of 0.05 R_s, thus reducing ϕ_ss until it agrees with ϕ_au for that rotation. Using this method we can find an optimum source surface distance ${r}_{\mathrm{ss}}^{{opt}}$ and an optimum flux density ${\phi }_{\mathrm{ss}}^{\mathrm{opt}}$, which closely agrees with ${\phi }_{\mathrm{au}}$ for that Carrington rotation. Figure 2(d) shows the perfect match between ${\phi }_{\mathrm{au}}^{\mathrm{EB}}$ and the corresponding ${\phi }_{\mathrm{ss}}^{\mathrm{opt}}$ for each rotation. Figure 2(e) shows the rotational values of r_ss^opt for ${\phi }_{\mathrm{au}}^{\mathrm{EB}}$, which vary from $1.05\,{R}_{s}$ in 2017 to $4.25\,{R}_{s}$ in 1987 for WSO_VSM and from $1.25\,{R}_{s}$ in 2009 to $4.35\,{R}_{s}$ in 1980 for MWO_VSM. Figure 2(e) also includes the 13-rotation running means of r_ss^opt for both ${\phi }_{\mathrm{au}}^{\mathrm{EB}}$ and ${\phi }_{\mathrm{au}}^{L}$, which both depict maxima around the three sunspot minima in 1976–1977, 1986–1987, 1996–1997 (and a smaller hump in 2006 and 2009 in ${\phi }_{\mathrm{au}}^{L}$), but no other systematic changes during the sunspot cycle.

The first two of these maxima are about 30%–50% above the mean level of r_ss^opt in the 1980s and 1990s.

The r_ss^opt values in Figure 2(e) depict an interesting long-term evolution. At the start of the time interval studied, in the late 1960s and early 1970s, the r_ss^opt values were slightly lower, about $1.8\,{R}_{s}$, than in the 1980s and 1990s. However, this is only based on MWO_VSM data and is therefore less certain than those later results which both MWO_VSM and WSO_VSM can verify. These two series agree on the considerably lower level of r_ss^opt values during the last 20 yr than in the three earlier decades. In fact, there is a step-like decrease in r_ss^opt after the sunspot minimum in 1996–1997, and a weakly decreasing trend thereafter. (Unfortunately, the WSO data is erroneous in the late 1990s over this time interval and is left out from Figure 2(e); see Section 2). Note that this decreasing trend in r_ss^opt during the last 20 yr is fairly linear, with only weak solar cycle related variations. Even the all-time minimum of ${\phi }_{\mathrm{au}}$ in 2009 (see Figure 2(d)) is very well reproduced by ${\phi }_{\mathrm{ss}}^{\mathrm{opt}}$ with only a small increase in the corresponding r_ss^opt.

Figure 3 shows one sample of a coronal field configuration during each of the four sunspot minima (in 1976, 1986, 1996, and 2008) using MWO_VSM data and the corresponding r_ss^opt values. Figure 3 shows that the structure of the coronal magnetic field has notably changed from the minimum in 1976 to the minimum in 2008. The coronal structure in 1976 depicts a large volume of closed magnetic field lines (magnetic loops), with a large fraction of those field lines connecting high northern latitudes with high southern latitudes. These long field lines extend quite far from the solar surface at the equator, indicating a large source surface distance. On the other hand, during the minimum in 2008 there are hardly any of such long field lines connecting the two hemispheres at high latitudes, and the closed field lines extend much lower in the corona than during the earlier minima.

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In this Letter we have shown that the coronal source surface distance has notably decreased from the mid-1970s until recently. This has also modified the topology of the coronal magnetic field so that the field lines now become radial at a lower altitude (smaller distance) than they did 20–50 yr ago. This shrinking of the coronal source surface from about 2.0 to $3.0\,{R}_{s}$ from the late 1970s until mid-1990s to about $1.5\,{R}_{s}$ in the last decade implies that the volume of the solar corona, where the magnetic field is the dominant form of energy, has reduced to less than one half during the last 20 yr. We find that a large fraction of this shrinking occurred as a step-like decrease in the late 1990s, in the early ascending phase of cycle 23. Similar abrupt changes roughly at the same time have been noted, e.g., in the distribution of sunspots (Lefèvre & Clette 2011; Clette & Lefèvre 2012), in sunspot magnetic field strength (Tapping & Valdés 2011; Watson et al. 2011), and in the relation between sunspot number and several solar EUV proxies (Lukianova & Mursula 2011; Livingston et al. 2012). Moreover, as seen in Figures 2(a), (b) and (d), the strength of the HMF, the open solar magnetic field, has significantly decreased over the same time interval (Wang et al. 2009; Hathaway & Rightmire 2010; Virtanen & Mursula 2019). These changes in the solar magnetic fields are most likely related to the decrease of the overall solar activity that mark the end of the exceptional period of the GMM of the twentieth century.

We acknowledge the financial support by the Academy of Finland to the ReSoLVE Centre of Excellence (project No. 307411). Wilcox Solar Observatory data used in this study were obtained via the website http://wso.stanford.edu courtesy of J.T. Hoeksema. This study includes data from the synoptic program at the 150 Foot Solar Tower of the Mt. Wilson Observatory, which is acknowledged. Data were acquired by SOLIS instruments operated by NISP/NSO/AURA/NSF.

Data used in this study was obtained from the following web sites:

WSO: http://wso.stanford.edu.

MWO: ftp://howard.astro.ucla.edu/pub/obs/synoptic_charts/fits/.

SOLIS/VSM: http://solis.nso.edu/0/vsm/vsm_maps.php.