1 Sources of the Solar Wind

Since the Skylab era of the 1970s, it has generally been agreed that high-speed (\(v > 500\) km s−1) solar wind originates from coronal holes, but the source(s) of the low-speed (\(v < 450\) km s−1) wind originally predicted by Eugene Parker (Parker, 1958) remain surprisingly controversial (Abbo et al., 2016). The prevailing scenario is that the solar wind is bimodal, with fast wind coming from coronal holes but slow wind coming from closed-field regions such as coronal streamers or active regions. This view is largely based on the distinctive properties of slow wind, including its greater spatiotemporal variability, higher ion charge-state ratios, enrichment in elements of low first-ionization potential (FIP), and lower He abundances near a solar minimum (Schwadron, Fisk, and Zurbuchen, 1999; Antiochos et al., 2011).

An alternative scenario is that both the fast solar wind and most of the slow wind come from coronal holes, with only a small fraction of the slow wind leaking out from coronal streamers in the immediate vicinity of the heliospheric current and plasma sheets. As will be discussed further, this view is supported by: (1) the empirical inverse correlation between wind speed and the rate of coronal flux-tube expansion (Wang and Sheeley, 1990; Arge and Pizzo, 2000; Poduval, 2016); (2) near-Earth observations of slow wind from small equatorial coronal holes near sunspot maximum (Wang and Ko, 2019); (3) in situ observations of slow wind with high Alfvénicity (D’Amicis and Bruno, 2015; Bale et al., 2019; D’Amicis, Matteini, and Bruno, 2019; Stansby, Horbury, and Matteini, 2019; D’Amicis et al., 2021).

We begin by discussing the wind from coronal streamers.

2 The Slow Streamer Wind

There are two basic types of coronal streamers: those that separate coronal holes of opposite magnetic polarity and have an associated current sheet, which we will call “helmet streamers”; and those that separate coronal holes of the same polarity and have no associated current sheet, which we will call “pseudostreamers” (see Figure 1). Both types of streamers have narrow plasma-sheet extensions; the helmet-streamer plasma sheet encloses the heliospheric current sheet (HCS).

Figure 1
figure 1

Two kinds of coronal streamers.

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The plasma-sheet extensions of helmet streamers and pseudostreamers mark the boundaries or separatrix surfaces between coronal holes of either polarity; this is sometimes called the “S-web” (Antiochos et al., 2011). As seen in sharpened coronagraph and eclipse images, the heliospheric plasma sheets (which are the source of the white-light emission in the outer K-corona) consist of narrow, ray-like structures (Figure 2) (Wang et al., 2000; Liewer et al., 2023). These rays might be formed by interchange reconnection between the outermost streamer loops and the adjacent open flux from coronal holes (Figure 3, top). This three-dimensional reconnection process would continually transfer streamer material to the plasma sheet, with each individual ray fading after all of the released material has transited the field of view.

Figure 2
figure 2

Sharpened image of the solar eclipse of March 29, 2006 (see Wang et al., 2007). The plasma-sheet extensions of the helmet streamers and pseudostreamers consist of narrow ray-like structures.

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Figure 3
figure 3

Top: Formation of a streamer ray by interchange reconnection. In the process, streamer material is released into the plasma sheet. Bottom: An expanding helmet-streamer loop pinches off; in 3D, a flux rope will be formed.

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The heliospheric plasma sheets may thus represent slow wind escaping from streamers. Another contribution to the slow streamer wind comes from the “blobs” that continually form at the cusps of helmet streamers. As shown by Solar Terrestrial Relations Observatory (STEREO) COR2A/B observations where the HCS is simultaneously viewed face-on and edge-on (Figure 4) (Sheeley et al., 2009), the blobs are actually flux ropes formed when the outward-expanding helmet-streamer loops reconnect with each other and pinch off (Wang et al., 2000; Chen et al., 2009; Lynch, 2020; Réville et al., 2022). This is analogous to what happens in coronal mass ejection (CME) eruptions, when the rising arcade loops reconnect with each other and pinch off to form flux ropes.

Figure 4
figure 4

White-light running-difference images showing “face-on” (left panels, COR2A) and “edge-on” (right panels, COR2B) views of helmet-streamer blobs. The face-on views, where the HCS lies in the sky plane, make it clear that blobs are flux ropes formed as the continually expanding helmet-streamer loops reconnect with each other and pinch off.

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In contrast to helmet streamers, pseudostreamers are relatively static and do not show a tendency to continually expand outward and pinch off to produce blobs. This is because the field that converges above their cusps is strong and unipolar, whereas the field above helmet streamers is weak and nonunipolar. Pseudostreamers and helmet streamers also give rise to fundamentally different kinds of CMEs (Wang, 2015): pseudostreamer CMEs are narrow and fan-shaped, whereas helmet streamer CMEs tend to be wide and bubble-shaped (making them more space-weather relevant). Again, this is because pseudostreamers are laterally confined by the like-polarity open flux on each side, whereas helmet streamers are surrounded by weak fields.

Helmet-streamer blobs are confined to the immediate vicinity of the HCS and accelerate gradually to speeds of ∼250 – 400 km s−1 by \(r\sim 30\) \(R_{\odot}\) (Figure 5). Their average velocity profile qualitatively resembles that of a 1 MK Parker isothermal wind; one possibility is that they are coupled by drag forces to the slow solar wind from just inside the boundaries of the adjacent coronal holes. As suggested by Large Angle and Spectrometric Coronagraph (LASCO) C3 images recorded during the 1997 activity minimum (Figure 6), the plasma sheet is only a few degrees wide and the blobs propagate along it like beads on a wire. The narrowness of the edge-on plasma sheet would be difficult to explain if the coronal hole–streamer boundary were as wide and ill-defined as sometimes postulated in models where interchange reconnection with streamer loops is the main source of slow solar wind (Rappazzo et al., 2012; Pontin and Wyper, 2015; Higginson et al., 2017). When the HCS lies in the sky plane, as sometimes occurs near sunspot maximum, low-speed wind is present at all position angles, permeated by blobs and streamer structure; this may have helped give rise to the concept of the “highly structured” slow solar wind, when in fact only a small component of the slow wind is being observed.

Figure 5
figure 5

Velocities of streamer blobs as a function of heliocentric distance (from Wang et al., 2000). Solid line shows the Parker 1 MK isothermal wind solution.

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Figure 6
figure 6

As illustrated by these SOHO/LASCO C3 images (left: polarized; right: running difference) recorded during the 1997 activity minimum, streamer blobs/flux ropes are confined to the immediate vicinity of the HCS/heliospheric plasma sheet, which is only a few degrees wide when seen edge-on.

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3 Slow Solar Wind from Coronal Holes

Comparisons between the solar-wind speed \(v_{\mathrm{E}}\) at 1 au and potential-field source-surface (PFSS) extrapolations of photospheric field maps suggest that \(v_{\mathrm{E}}\) is inversely correlated with the rate of flux-tube expansion at the Sun, parameterized by \(f_{\mathrm{ss}}\equiv (R_{\odot}/R_{\mathrm{ss}})^{2}(B_{0}/B_{\mathrm{ss}})\), where \(B_{0}\) is the field strength at the coronal base and \(B_{\mathrm{ss}}\) is the field strength at the source surface \(r = R_{\mathrm{ss}}\) (hereafter taken to be at 2.5 \(R_{\odot}\)). The rapidly expanding flux tubes that are predicted to give rise to low-speed wind are rooted just inside the boundaries of coronal holes and within the interiors of small holes. (Note that these regions are more likely to undergo rapid spatiotemporal variability than the interiors of large coronal holes, which contain the relatively slowly expanding flux tubes associated with high-speed wind.) However, at least in individual cases, the extrapolations are subject to large uncertainties, the most important source of which are the photospheric field measurements, which often do not adequately capture the weak (but critically important) dipole component and that may underestimate the flux in sunspots due to magnetograph saturation effects. Also, the current-free approximation may break down inside active regions, and the PFSS model does not include dynamical or time-dependent effects.

One way to lessen the dependence on the extrapolations is to track small equatorial coronal holes and see if we observe slow wind at Earth a few days after their central-meridian passage. Figure 7 (from Wang and Ko, 2019) shows an example of two small equatorial holes that produced low-speed wind at Earth during the 2014 sunspot maximum. As seen from the Fe xiv 21.1 nm synoptic map in the top panel, which was constructed from Solar Dynamics Observatory (SDO) Atmospheric Imaging Assembly (AIA) data, the holes are located in or at the borders of active regions; the corresponding PFSS-derived open-field regions (second panel) are characterized by large expansion factors (blue denotes values greater than 18). The in situ wind speeds from the two small holes range between ∼380 and ∼490 km s−1, while the O7+/O6+ ratios are on the order of 0.05 – 0.1, but increase steeply near the edges of both streams (third panel). Of particular interest is the Alfvénicity parameter \(C_{vB}\equiv (\delta \mathbf{v\cdot}\delta \mathbf{B})/ (\left | \delta v\right |\left |\delta B\right |)\) (bottom panel), which reaches values as high as 0.7 in both streams, comparable to those in high-speed wind. The Alfvénicity or cross-helicity measures the correlation between the transverse velocity and magnetic fluctuations, and is usually taken to be an indicator of coronal-hole origin. Recent Parker Solar Probe and Solar Orbiter measurements closer to the Sun have confirmed the Alfvénic nature of slow wind from small coronal holes (Bale et al., 2019; D’Amicis et al., 2021).

Figure 7
figure 7

Slow solar wind with high Alfvénicity observed from two small equatorial coronal holes. The PFSS-derived open-field areas in the second panel are color-coded to indicate the expansion factors: blue (\(f_{\mathrm{max}} > 18\)); green (\(9 < f_{\mathrm{max}} < 18\)); yellow (\(4 < f_{\mathrm{max}} < 9\)); orange (\(f_{\mathrm{max}} < 4\)). Here, \(f_{\mathrm{max}}\) denotes the maximum value of \(f(r) = (R_{\odot}/r)^{2}\left | B_{0}/B_{r}\right |\) (\(r\leq R_{\mathrm{ss}} = 2.5\) \(R_{\odot}\)) along a given flux tube; the maximum may occur at \(r < R_{\mathrm{ss}}\) in the vicinity of pseudostreamers. In the bottom two panels, the blue lines represent the IMF longitude angle \(\phi _{B}\), where \(\phi _{B} > 180^{\circ}\) (\(\phi _{B} < 180^{\circ}\)) corresponds to inward (outward) IMF. The O7+/O6+ measurements have an instrumental floor value of ∼0.05. The red diamonds in the bottom panel show the Alfvénicity parameter \(\left | C_{vB}\right |\), which reaches values as high as 0.7 in both slow-wind streams. The in situ data are plotted in time-reversed order and have been mapped back to the source surface assuming a transit time inversely proportional to the measured wind speed (black solid lines). From Wang and Ko (2019).

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There have been many studies of what is called “active-region wind” (Neugebauer et al., 2002; Liewer, Neugebauer, and Zurbuchen, 2004; Brooks, Ugarte-Urra, and Warren, 2015; Brooks et al., 2020; Stansby et al., 2021). As shown in Wang (2017), most of this wind probably originates from small coronal holes located inside or at the peripheries of active regions. Such holes are sometimes difficult to detect in extreme-ultraviolet (EUV) images because of the surrounding bright loops. Figure 8 provides an example of a small active-region hole that is barely visible in the 21.1 nm synoptic map but that produces slow wind with high Alfvénicity at Earth. If the hole is tracked as it rotates across the disk, it is found to be more clearly visible before it reaches the central meridian, when there is less bright material in the line of sight. It is sometimes assumed that coronal holes must have weak fields (Stansby et al., 2021), but EUV holes inside or near active regions have footpoint field strengths typically on the order of 30 G (Wang and Ko, 2019). Note also that large blue shifts inside active regions do not necessarily mean that the material is escaping into the heliosphere—active-region loops may be very dynamic, and their observed overdensities imply that they are not in hydrostatic equilibrium (Aschwanden et al., 2007).

Figure 8
figure 8

Small coronal holes inside or near active regions are sometimes partially hidden under the surrounding bright loops. The equatorial hole near longitude 135∘ is barely visible in the 21.1 nm synoptic map, but was clearly seen in individual images when located to the east of central meridian. From Wang and Ko (2019).

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PFSS extrapolations predict that most of the slow wind at Earth near solar minimum comes from just inside the polar-hole boundaries. Unlike the slow wind from the interiors of small coronal holes near solar maximum, most of this wind has low Alfvénicity, on the order of 0.1 – 0.2 (see Figure 9). This might suggest that we are actually observing streamer wind from just outside the polar-hole boundaries. However, in a study of Helios data recorded at 0.3 – 0.4 au during 1974 – 1978, Stansby, Horbury, and Matteini (2019) found that 80% of the wind was Alfvénic. Since this period was close to sunspot minimum, much of this wind probably came from just inside the polar holes. Stansby et al. divided the Alfvénic Helios wind into two categories: “isotropic Alfvénic,” having speeds ∼200 – 500 km s−1, low perpendicular proton temperatures \(T_{\mathrm{p}\perp}\), and high charge-state ratios; and “anisotropic Alfvénic,” having speeds ∼300 – 700 km s−1, high \(T_{\mathrm{p}\perp}\), and low charge-state ratios. They attributed the isotropic Alfvénic wind to coronal-hole boundaries and the anisotropic Alfvénic wind to coronal-hole cores. (The remaining ∼20% of the Helios wind was described as “isotropic non-Alfvénic” and attributed to “small-scale transients.”)

Figure 9
figure 9

PFSS extrapolations predict that most of the slow wind near sunspot minimum comes from just inside the polar-hole boundaries. This wind has low Alfvénicity (∼0.1 – 0.2) at 1 au, unlike the slow wind from small coronal holes.

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The simplest way to explain these different observations is to assume that, except in the immediate vicinity of the heliospheric plasma sheets, all of the solar wind is Alfvénic near the Sun, but the Alfvénicity of the wind from just inside coronal-hole boundaries is reduced by propagation effects beyond ∼0.5 au. Such effects might include shear flow/Kelvin–Helmholtz or compressional interactions with neighboring wind streams or with the heliospheric plasma sheets (Roberts et al., 1987, 1991). It should be noted that the flux tubes rooted in the interiors of small coronal holes continue to expand superradially until ∼10 \(R_{\odot}\), so most of this wind is not subject to boundary interactions. In contrast, the rapidly expanding flux tubes rooted near the polar-hole boundary reconverge beyond the helmet-streamer cusp, so this wind remains close to the boundary far from the Sun.

4 Relating Solar-Wind Properties to the Coronal Magnetic Field

Figure 10 displays scatter plots constructed from near-Earth OMNI data, Advanced Composition Explorer (ACE) Solar Wind Ion Composition Spectrometer measurements, and PFSS-derived coronal-field parameters (see Wang, 2016). The blue points represent 8-h averages of data from 1998 – 2004 (near sunspot maximum), while the red points represent similar averages for the period 2007 – 2009 (near sunspot minimum).

Figure 10
figure 10

Scatter plots relating solar-wind parameters to the coronal magnetic field. Blue points represent the period 1998 – 2004 (near sunspot maximum), while red points represent the period 2007 – 2009 (near sunspot minimum). Note that: (1) for any given wind speed, the O7+/O6+ ratios are higher near sunspot maximum than near sunspot minimum (panel (a)); (2) the wind speed tends to decrease with increasing flux-tube expansion factor \(f_{\mathrm{ss}}\), independent of solar-cycle phase (panel (b)); (3) there is no systematic relationship between the wind speed and the footpoint field strength \(B_{0}\) (panel (c)); (4) the mass and energy flux densities at the coronal base increase roughly linearly with \(B_{0}\) (panels (e) and (f)). From Wang (2016).

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In Figure 10(a), the solar wind speed is plotted against the O7+/O6+ ratio. For any given wind speed, the O7+/O6+ ratio is seen to be larger near sunspot maximum than near sunspot minimum. This is because the oxygen charge-state ratio is highly correlated with the footpoint field strength \(B_{0}\): as the median value of \(B_{0}\) decreased from 21 G during 1998 – 2004 to 3 G during 2007 – 2009, the median value of O7+/O6+ decreased from 0.15 to only 0.06. Clearly, the oxygen charge-state ratio alone cannot be used to distinguish between coronal hole and streamer wind, as is sometimes assumed (see, e.g., Zhao, Zurbuchen, and Fisk, 2009).

In Figure 10(b), the wind speed is plotted against the flux-tube expansion factor \(f_{\mathrm{ss}}\). Independent of the solar-cycle phase, the wind speed tends to decrease as \(f_{\mathrm{ss}}\) increases. This is because, along rapidly expanding flux tubes, the magnetic energy dissipation is concentrated at low heights, leading to higher mass flux densities but less energy per proton (Wang, Ko, and Grappin, 2009) (cf. Leer and Holzer, 1980, who showed that energy addition in the subsonic region increases the mass flux, whereas energy addition in the supersonic region increases the wind speed).Footnote 1 The higher mass flux densities at the base of the slow wind imply stronger chromospheric evaporation (higher enthalpy fluxes), which may act to drag out low-FIP elements.

The empirical \(v_{\mathrm{E}}\)–\(f_{\mathrm{ss}}\) relationship in Figure 10(b) exhibits a great amount of scatter, particularly at low wind speeds. The main source of this scatter is likely to be the photospheric field maps used to trace back to the solar-wind footpoints; especially critical in this respect are the polar fields (and, more generally, the lowest-order multipoles of the photospheric field), which are often poorly measured. This means that the source regions are frequently misidentified. (Another factor contributing to the scatter is the very rapid change in \(f_{\mathrm{ss}}\) just inside the boundaries of the polar holes, where most of the slow wind near solar minimum originates.) Thus, when using \(f_{\mathrm{ss}}\) to forecast the solar wind speed, it is essential first to verify that the predicted open-field regions correspond to observed coronal holes; otherwise, there is no reason to expect the forecast to be correct. In addition, of course, the acceleration of the solar wind is likely to depend (if less strongly) on many other physical parameters. For example, open flux tubes located near pseudostreamer boundaries will undergo rapid divergence at low heights followed by reconvergence above the pseudostreamer null point; this nonmonotonic expansion is not described by \(f_{\mathrm{ss}}\) alone.

Figure 10(c) shows that the wind speed is independent of \(B_{0}\) itself: a given footpoint field strength may have any wind speed associated with it. While the wind speed tends to increase with the source-surface field strength (Figure 10(d)), the value of \(\vert B_{\mathrm{ss}}\vert \) associated with a given wind speed is higher near sunspot maximum than near sunspot minimum.

Figures 10(e) and (f) show that the mass and energy flux densities at the coronal base increase almost linearly with the footpoint field strength. This important (but often overlooked) result was obtained by applying mass and energy conservation along an Earth-directed flux tube, using OMNI measurements of the wind speed, proton density, and radial IMF strength \(B_{\mathrm{E}}\), and setting the in situ solar-wind energy-flux density to \((1/2)\rho _{\mathrm{E}}v_{\mathrm{E}}^{3}\) (where \(\rho _{\mathrm{E}}\) denotes the proton mass density). Note that, because both the mass-flux density at the coronal base and the net areal expansion of a flux tube scale as \(B_{0}\), the mass-flux density at 1 au is roughly constant: \(\rho _{\mathrm{E}}v_{\mathrm{E}} = (B_{\mathrm{E}}/B_{0})\rho _{0}v_{0}\propto B_{ \mathrm{E}}\).

5 The Role of Coronal Plumes in the Solar Wind

Skylab EUV spectroheliograms already showed that coronal/polar plumes were located above bright points and compact network brightenings containing minority-polarity flux, suggesting that they were energized by interchange reconnection with small bipoles (Wang and Sheeley, 1995); this inference was supported by subsequent SOHO Extreme-ultraviolet Imaging Telescope observations. Until recently, plumes were thought to be a minor constituent of coronal holes and the solar wind (see the review of Poletto, 2015). However, higher-sensitivity EUV observations, such as the SDO/AIA 17.1 nm image displayed in Figure 11, now make it apparent that diffuse plume emission overlies every network concentration inside a coronal hole.

Figure 11
figure 11

SDO/AIA and HMI images showing that 17.1 nm plume emission overlies every network concentration inside a coronal hole. Earlier instruments might have detected only the bright plumes near the top-left and bottom-right corners of the Fe ix image. From Wang (2022).

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Comparisons between sequences of AIA and Helioseismic and Magnetic Imager (HMI) images show that plumes form as supergranular flows converge and concentrate the network and internetwork flux (including any embedded minority-polarity flux) into dense clumps, and fade as the flows diverge and the clumps are dispersed (Figure 12) (Wang, Warren, and Muglach, 2016; Avallone et al., 2018; Qi et al., 2019). The associated timescales of several hours to a day are similar to those of solar-wind microstreams (Neugebauer et al., 1995; Kumar et al., 2023). At the same time, granular motions on timescales of ∼10 min or less trigger smaller-scale reconnection events within the converging and diverging network concentrations, giving rise to individual “jetlets” (Raouafi and Stenborg, 2014; Kumar et al., 2022). The ohmic heating accompanying the interchange reconnection also evaporates material from below to form a diffuse cloud permeated by the cluster of jetlets. Jets/jetlets are also present throughout the network outside of plumes; because the fields are weaker and the heating less concentrated there, the overlying densities are lower, the EUV emission is fainter, and the jets are less closely packed.

Figure 12
figure 12

Coronal plumes form as converging supergranular flows bring network and internetwork flux together to form dense clumps, and fade as the flows diverge and the clumps are dispersed. The timescales of several hours to a day are on the same order as those of long-period Alfvén waves and microstreams. From Wang, Warren, and Muglach (2016).

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Most models for the turbulent dissipation of Alfvén waves require the presence of reflected waves that interact with the outgoing waves (Dmitruk, Milano, and Matthaeus, 2001; Dmitruk et al., 2002; Cranmer and van Ballegooijen, 2005). However, for an Alfvén wave to be reflected, its period should satisfy \(P_{\mathrm{wave}} > P_{\mathrm{crit}}\simeq 2\pi \left |\partial v_{\mathrm{A}}/ \partial r\right |^{-1}\sim 1\) h. This criterion may be satisfied by waves generated during the formation of plumes through supergranular motions, but not by waves associated with granular motions and jetlets. Using the upgraded Coronal Multi-channel Polarimeter, Sharma and Morton (2023) found that the perpendicular correlation length for Alfvén waves in the lower corona peaked near 10 Mm, which is comparable to a supergranule radius.

6 Evidence That Magnetograms Underestimate the Amount of Minority-Polarity Flux Inside Coronal Holes and Other “Unipolar” Regions

The idea that coronal holes and the solar wind are energized by interchange reconnection with small-scale flux (or the “magnetic carpet”) has long been discounted, because magnetograms show very little minority-polarity flux inside coronal holes and other “unipolar” regions (such as active-region plages). Thus, Newkirk and Harvey (1968) and DeForest et al. (1997) found that polar plumes were located above purely unipolar network concentrations, contrary to the conclusion of Wang and Sheeley (1995). However, comparisons of EUV images with HMI magnetograms from SDO often show large amounts of loop-like fine structure inside areas that are purely unipolar according to the corresponding magnetograms. As a typical example, Figure 13 displays an Fe ix 17.1 nm plume inside a coronal hole. As may be seen from the sharpened image in the middle panel, the core of the plume contains a cluster of small loops with sizes of around 5 Mm; but the corresponding magnetogram (right-hand panel), which has 0.5′′ pixels, shows almost no underlying minority-polarity flux. Similarly, AIA 17.1 and 19.3 nm images of plages often show small loops around the footpoints of active-region loops without corresponding minority-polarity signatures in simultaneous HMI magnetograms.

Figure 13
figure 13

The cluster of small loops in the core of this plume overlies a purely unipolar flux concentration. The loops have sizes of order 5 Mm, greatly exceeding the HMI 0.5′′ pixel size. Adapted from Wang (2022).

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7 The Rate of Small-Scale Flux Emergence Inside Coronal Holes

In a study using SOHO Michelson Doppler Imager (MDI) magnetograms, Hagenaar, DeRosa, and Schrijver (2008) found that the flux-emergence rate \(E_{\mathrm{ER}}\) of ephemeral regions (ERs: small bipoles with fluxes ≲1020 Mx) depended on a flux-imbalance parameter \(\xi \), which was equal to 1 in purely unipolar regions and equal to 0 in the quiet Sun: \(E_{\mathrm{ER}} = (1.075 - 0.793\xi ^{2})\times 10^{-3} \phantom{.} {\mathrm{Mx}} \phantom{.} {\mathrm{cm}}^{-2} \phantom{.} {\mathrm{s}}^{-1}\). We shall assume instead that the emergence rate is the same in coronal holes as in the quiet Sun, and set \(\xi = 0\). In addition, we scale the MDI fluxes upward by a factor of 1.7 to correct for magnetograph saturation (Tran et al., 2005) and allow for the contribution of ERs with fluxes down to 1018 Mx, corresponding to sizes ∼2 Mm, the approximate height of the coronal base. The emergence rate is then \(E_{\mathrm{ER}}\simeq 2.25\times 10^{-3} \phantom{.} {\mathrm{Mx}} \phantom{.} {\mathrm{cm}}^{-2} \phantom{.} {\mathrm{s}}^{-1}\).

8 Estimating the Energy-Flux Density from Interchange Reconnection

To estimate the contribution of ERs to coronal-hole heating, let \(B_{0}\) denote the average strength of the open flux at the coronal base. The timescale for an equal amount of minority-polarity flux to emerge into the corona is then

$$ \tau _{\mathrm{recyc}}\sim \frac{B_{0}}{E_{\mathrm{ER}}(\Phi _{\mathrm{ER}}\gtrsim 1\times 10^{18}\phantom{.}{\mathrm{Mx}})/2} \sim 2.5\left (\frac{B_{0}}{10\phantom{.}{\mathrm{G}}} \right )\left [ \frac{2.25\times 10^{-3}\phantom{.}{\mathrm{Mx}} \phantom{.}{\mathrm{cm}}^{-2}\phantom{.}{\mathrm{s}}^{-1}}{E_{\mathrm{ER}}(\Phi _{\mathrm{ER}} \gtrsim 1\times 10^{18}\phantom{.}{\mathrm{Mx}})}\right ] \phantom{.} {\mathrm{h}}, $$
(1)

where we note that only one-half of the ER flux has polarity opposite to that of the open flux. Over this “recycling” or flux-replacement time, all of the open flux will have undergone interchange reconnection, with magnetic energy being dissipated in a layer whose thickness is determined by the heights of the ER loops. Denoting the average loop height by \(\langle h_{\mathrm{ER}}\rangle \), we obtain for the energy-flux density arising from interchange reconnection (Wang, 2020):

$$\begin{aligned} F_{\mathrm{ER0}}&\sim \frac{B_{0}^{2}}{8\pi}\left ( \frac{\langle h_{\mathrm{ER}}\rangle}{\tau _{\mathrm{recyc}}}\right ) \\ &\sim 2.9\times 10^{5} \phantom{.} \left (\frac{B_{0}}{10\phantom{.}{\mathrm{G}}}\right )\left ( \frac{\langle h_{\mathrm{ER}}\rangle}{6.4\phantom{.}{\mathrm{Mm}}}\right )\left [ \frac{E_{\mathrm{ER}}(\Phi _{\mathrm{ER}}\gtrsim 1\times 10^{18}\phantom{.}{\mathrm{Mx}})}{2.25\times 10^{-3}\phantom{.}{\mathrm{Mx}} \phantom{.}{\mathrm{cm}}^{-2}\phantom{.}{\mathrm{s}}^{-1}}\right ] \phantom{.} {\mathrm{erg}} \phantom{.} {\mathrm{cm}}^{-2} \phantom{.} {\mathrm{s}}^{-1}. \end{aligned}$$
(2)

Here, \(\langle h_{\mathrm{ER}}\rangle \) has been normalized to 6 Mm, corresponding to the size of a bipole with total flux \(\Phi _{\mathrm{ER}}\sim 1\times 10^{19}\) Mx.

This estimate of the energy-flux density inside coronal holes is close to the value given by Withbroe and Noyes (1977). The scaling \(F_{\mathrm{ER0}}\propto B_{0}\) also agrees with the empirical result derived from energy conservation along an Earth-directed flux tube (Figure 10(f)). Note that setting \(B_{0}\sim 300\) G, the typical value for an active-region plage, would give an energy-flux density on the order of 107 erg cm−2 s−1, enough to heat the active-region corona (Wang, 2022). Also, as indicated by Figure 12, the field strength in the core of a plume may be as high as ∼200 G; much of the implied extra heating goes into increasing the local electron density and the radiative loss rate, making the plume visible in white light and the EUV.

9 Concluding Remarks

We have presented evidence that both the fast solar wind and the bulk of the slow wind come from coronal holes, and that coronal holes (and the solar corona in general) are heated by interchange reconnection with small-scale flux. The footpoint reconnection occurs just above the coronal base and gives rise to ohmic dissipation, small jets, and Alfvén waves (Kigure et al., 2010). The heat is conducted both upward into the corona and downward through the transition region, with chromospheric evaporation being the main driver of the solar-wind mass flux (as well as accounting for the overdensity of active-region loops). The wind speed is largely determined by the rate of flux-tube expansion (see Figure 14). Along rapidly expanding flux tubes, the outward conductive heat-flux density falls off rapidly, as does the temperature; the Alfvén speed also falls off rapidly, leading to less wave dissipation at greater heights in models where the turbulent dissipation rate increases with the local Alfvén speed (see, e.g., Dmitruk, Milano, and Matthaeus, 2001; Dmitruk et al., 2002); both of these effects act to give slow wind. Conversely, along slowly expanding flux tubes, the temperature and wave dissipation rate fall off slowly, leading to fast wind. Note also that the reflection of Alfvén waves beyond the sonic point will impart momentum to the outflow.

Figure 14
figure 14

The rate of flux-tube expansion determines the distribution of the heating and thus the wind speed.

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We were led to the conclusion that the solar wind is energized by interchange reconnection with small-scale flux after noting a clear discrepancy between EUV images and magnetograms (see Wang, 2020, and references therein). Other investigators have recently reached similar conclusions based on 17.1/17.4 nm images showing a forest of jetlets above the polar limb (Raouafi et al., 2023; Chitta et al., 2023) or on in situ observations of magnetic switchbacks, interpreted as flux ropes injected during the interchange reconnection process (Drake et al., 2021; Bale et al., 2023). We remark that the newly reconnected open-field lines will have curved bases, but these will rapidly straighten out due to the effect of magnetic tension and will not themselves propagate outward in the form of switchbacks. However, as shown by Squire, Chandran, and Meyrand (2020), some of the Alfvén waves generated in the process may subsequently evolve nonlinearly into a turbulent state containing local reversals in the radial field component.

It should be emphasized that the solar wind is not simply a collection of jets/jetlets; their observed speeds are only on the order of 100 km s−1, well below the escape velocity, and they are not themselves the dominant energy or momentum source of the wind. Their main significance is that they indicate that reconnection is occurring everywhere at the base of the solar wind, accompanied by strong localized heating, chromospheric evaporation, and the generation of MHD waves.

That the solar wind is driven by multiple, small-scale reconnection events near the coronal base may have implications for the turbulence power spectrum of the interplanetary field, which is characterized by a \(1/f\) dependence at low frequencies but falls off as \(\sim f^{-5/3}\) in the higher-frequency inertial range (see, e.g., Matthaeus and Goldstein, 1986; Huang et al., 2023). According to one interpretation (Matthaeus and Goldstein, 1986; Mullan, 1990), the \(1/f\) component originates from an inverse-cascade process involving the coalescence of relatively small-scale magnetic structures near the Sun. This would be consistent with the merging of the individual reconnection/evaporative outflows as they propagate outward from the coronal base, with the merging process being driven by the underlying converging supergranular flows, by transverse pressure gradients, and by shear-flow interactions. We note that, in the presence of the relatively strong (\(\beta < 1\)) coronal field, turbulent flows such as those excited by Kelvin–Helmholtz instabilities will have a quasi-2D nature, which in turn may lead to the coalescence of vortices (inverse cascades). As the field weakens at greater distances, the turbulence will become fully 3D and the direction of the cascades will reverse, resulting in the standard \(f^{-5/3}\) Kolmogorov behavior.