Early spacecraft data in the 1960s revealed solar wind properties, which could not be well explained by models in which the electron pressure gradient was the principal accelerating force. The Alfvén waves discovered around 1970 were thought for a while to provide additional energy and momentum, but they ultimately failed to explain the rapid acceleration of the fast wind close to the Sun. By the late 1970s, various data were suggesting the importance of the ion-cyclotron resonance far from the Sun. This notion was soon applied to the acceleration region close to the Sun. The models, which resulted, suggested that the fast wind could be driven mainly by the proton pressure gradient. Since the mid-1990s, Solar and Heliospheric Observatory has provided remarkable data, which have verified some of the predictions of these theories, and given impetus to studies of the ion-cyclotron resonance as the principal mechanism for heating the coronal holes, and ultimately driving the fast wind. After a historical review, we discuss the basic ideas behind current research, emphasizing the particle kinetics. We discuss remaining problems, especially the source of the ion-cyclotron resonant waves.
Since its launch in December 1995, the Solar and Heliospheric Observatory (SOHO) has yielded remarkable data concerning the origin, acceleration and heating of the solar wind; see especially Cranmer (2002, 2004) and references therein. These data have also impacted our ideas concerning the long-standing problems of what heats the solar corona and chromosphere. (The coronal heating problem is certainly the most famous, but it is often forgotten that the power requirements of the chromosphere, per unit of solar surface area, are comparable to, and sometimes greater than, those of the corona.) SOHO has forced us to discard some long-held ideas, and to revive others, which had been prematurely dismissed. The SOHO data have also stimulated a number of entirely new ideas, which are still in formative stages. The purpose of this review is to survey our current understanding of the drivers of the fast solar wind. Since we may be at a watershed, we will also review the historical context, showing how current ideas differ from, and bear relation to, what has gone before. In short, we will try to show how we got to where we are today.
We will for the most part concentrate on the fast solar wind, for three reasons: first, we have a pretty good idea where it comes from, in particular the large coronal holes, which are usually present at the poles (except around the time of solar maximum), and which are especially dominant near solar minimum; see Miralles et al. (2002, 2004) for discussions of solar wind flows out of smaller non-polar holes. Second, the fast wind seems simpler. It is much steadier than the slow wind, and seems to be less structured. This was especially apparent in the Ulysses data (e.g. Phillips et al. 1995). It has always been hoped that the apparently simpler fast wind would be more amenable to yielding its secrets, but this remains to be seen. Third, and perhaps most importantly, the fast wind has long been known to be less influenced by Coulomb collisions than the slow wind (e.g. Neugebauer 1981). Though collisions cannot be entirely neglected, this means that signatures of other kinetic processes, such as wave–particle interactions, are more readily seen in the data (both proxy data such as from SOHO, and in situ particles and fields data from a wealth of spacecraft). Indeed, one of the conclusions of this paper will be that microscale kinetic processes are an essential ingredient of solar wind physics; magnetohydrodynamics (MHD) cannot provide the whole story, unless the microscale processes can be correctly built into the MHD equations via heating and acceleration terms, transport coefficients, etc.
The plan of this paper is as follows: §2 briefly reviews the electron-driven wind originated by Parker (1958), and its shortcomings. The successor to the electron-driven wind was the wave-driven wind, which is reviewed in §3, along with its shortcomings. Section 4 discusses a variation of the wave-driven wind, which we refer to as the proton-driven wind; this section also discusses the SOHO data, which gave the first support to this idea. Ion-cyclotron resonances were part of the theoretical development of the proton-driven wind, and their importance is in fact suggested by the data from SOHO. Section 5 discusses, from a simple physical point of view, how the resonances can explain, at least qualitatively, the observations. Possible sources for the ion-cyclotron waves are reviewed in §6. A few promising recent ideas related to the ion-cyclotron picture are mentioned in §7. The paper concludes with a view to the future in §8.
The reader should be aware that we are not attempting a complete literature survey. We do try, however, to reference key papers, which themselves provide references to the bulk of the literature. See especially recent reviews by Hollweg & Isenberg (2002; hereafter ‘HI’) and Cranmer (2002, 2004).
2. The electron-driven wind
Parker (1958) noted that the thermal conductivity of ionized hydrogen scales as , where T is temperature and subscript ‘e’ denotes electrons. If outward thermal conduction dominates the heat equation, then Te∝r−2/7 in a spherically symmetric corona, r being heliocentric distance. For a corona in static equilibrium, Parker showed that this slow decline of Te leads to an asymptotic plasma pressure at r→∞ which is many orders of magnitude larger than the interstellar pressure. Parker concluded that with nothing to contain the asymptotic pressure, the corona must expand. In essence, the expansion is driven by the electron pressure gradient, because Te remains high (declining only as r−2/7).
However, in a later review, Parker (1965) compared detailed theoretical predictions with the by then known properties of the solar wind at 1 AU. He concluded ‘Thus the model for the hypothetical conduction corona leads to a temperature falling too rapidly with radial distance from the Sun, indicating that the actual solar corona is probably actively heated for some considerable distance by the dissipation of waves.’ The jury is still out on whether waves are responsible, but we now know that he was right about extended coronal heating.
Parker's point was driven home by Hartle & Sturrock (1968), who presented the first two-fluid model of the solar wind, with separate energy equations and unequal temperatures for electrons and protons. They obtained a very slow wind: 250 km s−1 at 1 AU compared to a typical fast wind speed of 750 km s−1 (see Feldman et al. 1976 for an early review of the properties of the fast wind). They also found that the model protons were much too cold: 4400 K at 1 AU compared to several times 105 K in fast wind. They concluded that ‘departures of the solar wind characteristics near Earth from those of this model are also to be attributed to heating by a flux of non-thermal energy.’
Before concluding this section, we mention that neither Parker nor Hartle & Sturrock said anything about coronal heating. They began their models at r=rS with 106 K coronal temperatures, and no subsequent heat addition (rS is the solar radius).
3. The wave-driven wind
The discovery of the ubiquitous presence of Alfvén waves in the near-Earth solar wind was the next major advance. Belcher & Davis (1971) showed that magnetic and velocity fluctuations in the solar wind closely obey the correlation expected for non-compressive Alfvén waves (sometimes called the ‘intermediate mode’). From the sign of the correlation, and the known direction of the background magnetic field B0, they were able to deduce that the waves predominantly propagate away from the Sun; to the extent that the waves are purely Alfvénic, this means that the time-averaged Poynting flux is along B0 in the outward sense, as viewed in the frame moving with the wind. The waves have large amplitudes (|δB|/B0≈1, where the prefix ‘δ’ denotes the wave fluctuation and the subscript ‘0’ denotes the background), but they are nonetheless weakly compressive (e.g. δ|B|/B0≪1 and δρ/ρ0≪1, where ρ is the plasma density). Most of the wave power resides at long periods, of the order of hours. The waves are most purely Alfvénic and outward-propagating in the fast wind.
The predominant outward propagation strongly suggests that the Sun is the source of these waves. (It is fair to say that the Sun being a source of waves is not a recent discovery. It should be kept in mind that the direct solar observations of waves have not yet detected long-period waves, which seem to dominate in the solar wind, and, as we shall see, rather close to the Sun as well.)
It was immediately realized that the Alfvén waves might be Parker's ‘waves’ and Hartle & Sturrock's ‘flux of non-thermal energy.’ Belcher (1971) and Alazraki & Couturier (1971) inaugurated the concept of the wave-driven wind by noting that the waves exert a volumetric force −∇〈δB2〉/8π on the wind (in cgs units, which we shall use throughout); the angle brackets denote a time-average. This force can be thought of as the gradient of the radiation pressure. Hollweg (1973) showed how the wave energy equation could be extended to include dissipation and plasma heating. With heating and wave pressure, the wave-driven models had the potential to explain the high speeds and hot protons observed in the fast wind far from the Sun. And indeed they did (e.g. Hollweg 1978).
Though these models generally succeeded in explaining data far from the Sun, they failed close to the Sun. New coronal hole density data, obtained from spacecraft in the early 1990s, verified previous evidence that the density declines very rapidly with increasing r (Guhathakurta & Holzer 1994; Fisher & Guhathakurta 1995; Guhathakurta & Fisher 1998). That, combined with mass conservation requires the flow speed to increase very rapidly with r, reaching speeds in excess of 200 km s−1 already at r=3rS. The wave-driven models could not achieve such rapid accelerations. The reason is simply that the wave pressure, 〈δB2〉/8π, becomes comparable to other terms in the momentum balance only at greater values of r, in excess of 5rS or so (Hollweg 1978). (This does not mean, however, that the wave pressure acting in r>5rS does not contribute substantially to the final solar wind speed far from the Sun.)
There were other difficulties as well, particularly concerning heavy ions, the best observed being He++ because of its relatively high abundance. The ions flow faster than the protons, roughly by the Alfvén speed VA, and they are hotter than the protons, roughly in proportion to their masses (see Neugebauer 1992 for a review). (These properties of the ions are most noticeable in the fast wind, because Coulomb collisions are less important there.) Efforts (mainly in the early 1980s) to explain these observations generally invoked the ion-cyclotron resonance, as a means of both accelerating the flows and increasing the temperatures (see reviews by Isenberg 1983; Cranmer 2002, 2004; and HI). These models were only partially successful. One difficulty was that once an ion acquires a sufficient flow speed, it will drop out of resonance so that further acceleration and heating cease. The other difficulty was that these early studies considered resonant acceleration and heating only far from the Sun, well beyond the region where most of the solar wind acceleration had already taken place. We will soon see that the SOHO observations show extensive heavy ion heating close to the Sun in the acceleration region.
There were other indications that the ion-cyclotron resonance was at work. As reviewed by Marsch (1991), in situ measurements of proton distribution functions often show that the protons in the vicinity of the peak of the distribution are anisotropic, with more thermal energy perpendicular to the local magnetic field than along the field; the anisotropies are most noticeable in high-speed wind, presumably because Coulomb collisions are less important there. Moreover, the average magnetic moment, Tp⊥/B, increases with distance from the Sun (Tp⊥ is the proton temperature perpendicular to the magnetic field and B is the field strength). Both of these observations suggest that perpendicular heating is occurring in interplanetary space, and that in turn suggests the cyclotron resonance.
4. The proton-driven wind
Hollweg (1986) and Hollweg & Johnson (1988; hereafter ‘HJ’) applied these ideas to the solar wind close to the Sun. They assumed that the Sun launches low-frequency Alfvén waves which undergo a turbulent cascade to high frequencies, where they are dissipated into heat via the cyclotron resonance. HJ assumed that the resonant dissipation would heat only the protons, and that this was the only source of coronal heating. The heating rate was dictated by the rate at which energy cascades to high frequencies; HJ took the Kolmogorov rate, (Q is the volumetric heating rate, ρ is plasma density, 〈δV2〉 is the velocity variance associated with the waves and Lcorr,⊥ is the correlation length perpendicular to the magnetic field). These models also included acceleration by the wave pressure. The waves were taken to be outward-propagating in the short-wavelength limit. (We shall later see that this is formally inconsistent with turbulence.)
These models succeeded in reproducing the observed high-speed wind far from the Sun, as well as the rapid acceleration close to the Sun. HJ found that the protons close to the Sun, in r>3rS, were considerably hotter than the electrons. The pressure of the hot protons was mainly responsible for the rapid flow acceleration. Unfortunately, the available data at the time indicated that the protons were not hot close to the Sun; these models were discarded.
Isenberg (1990) extended HJ by including He++. (Quasilinear kinetic theory was used to apportion the cascaded energy between protons and helium.) It was hoped that the helium would absorb enough energy so that the protons would not get too hot. This did not turn out to be the case, and the model was discarded.
As it happened, the proton data which led to the dismissal of these models was misleading. HJ and Isenberg (1990) actually anticipated the results, which SOHO would soon obtain for coronal holes: hot coronal protons, and heavy ions that flow faster and are more than mass-proportionally hotter than the protons close to the Sun. Moreover, the SOHO results, especially from the Ultraviolet Coronagraph Spectrometer (UVCS), suggest that protons and ions in coronal holes close to the Sun are heated mainly perpendicular to the magnetic field; this result is firm for O+5 (Dodero et al. 1998; Kohl et al. 1998; Antonucci et al. 2000), but less certain for protons.
Thus, the SOHO/UVCS results suggest: (i) consistent with the perpendicular heating of O+5, the cyclotron resonance is at work. (ii) With protons hotter than electrons, the solar wind flow is mainly proton-driven, via the divergence of the (presumably anisotropic) proton pressure tensor. (iii) At least in the fast wind, coronal heating is proton and ion-dominated. It is not Joule dissipation. Do these conclusions apply to other coronal regions as well? (iv) The peculiar properties of heavy ions, viz. their high temperatures and fast flow speeds, originate close to the Sun in the acceleration region of the wind, not far from the Sun as had been assumed previously. (v) Coronal heating extends well into the acceleration region and into the supersonic wind. Most remarkable is O+5, which has a temperature of 3×108 K at r=3.5rS; the temperature is still an increasing function of r, in spite of strong adiabatic cooling (Kohl et al. 1998). Coronal heating and solar wind acceleration must be treated together. This stands in strong contrast to many early studies (e.g. Parker 1958; Hartle & Sturrock 1968), which assumed that all coronal heating took place in a thin layer at the coronal base.
There is, however, one note of caution. The proton and oxygen temperatures are derived from UVCS spectral line widths, assuming that non-thermal broadening due to waves or turbulence is negligible. This is almost certainly true for O+5, which shows considerably higher line widths than the protons; recall that E×B drifts would be the same for all particles if their flow speeds differ by much less than VA (E is the electric field). Esser et al. (1999) and Cranmer (2004) have used this approach to constrain the wave amplitudes. Other constraints rely on modelling assuming that the entire solar wind energy is provided by Alfvén waves (e.g. HI); these models suggest that waves or turbulence do not contribute substantially to any of the observed line widths.
Other instruments on SOHO, especially SUMER (Solar Ultraviolet Measurements of Emitted Radiation), also show indications of ion heating low in the corona, in regions where particle collisions are much more important than in the regions investigated by UVCS (e.g. Tu et al. 1998; Moran 2003; Peter & Vocks 2003).
Before closing this section, we mention that other authors investigated the effects of hot coronal protons (Esser & Habbal 1995, 1996; Hansteen & Leer 1995; McKenzie et al. 1995, 1997; Axford & McKenzie 1996; Esser et al. 1997; Lie-Svendsen et al. 2001), but only ad hoc heating functions were used. Li (1999, 2002), Li & Habbal (1999) and Li et al. (1999, 2004) have extended the original models of HJ and Isenberg to include thermal anisotropy, wave dispersion and other effects.
5. The ion-cyclotron resonance
It is well known that the cyclotron resonance occurs when the Doppler-shifted wave frequency seen by a particle matches the particle's cyclotron gyration frequency. Formally, the resonance condition is(5.1)where ω is the wave angular frequency, k∥ is the wavenumber along B, V∥ is the particle's drift speed along B and Ω is the gyro-frequency. The ‘±’ sign takes into account the sense of rotation of the wave electric field vector; for ions resonating with left-hand polarized waves the ‘+’ sign is appropriate, while the ‘−’ sign would be used for right-hand waves. When equation (5.1) is satisfied, the gyrating particle sees a constant electric field and the particle's energy changes secularly. It is sometimes forgotten, however, that the secular energy change can be a gain or a loss, depending on the phase of the particle's gyro-motion relative to the wave. (One could build a cyclotron to decelerate particles, though it is hard to imagine why one would want to do so.) In a random field, that phase will be random, the particle will gain or lose energy randomly, and, consequently, the particle will undergo a random walk in velocity space. Since random walks lead to diffusion, resonance in a random wave will lead to diffusion in velocity space, frequently referred to as pitch-angle diffusion; heating and acceleration of a particle distribution is formally analysed using diffusion equations in velocity space.
However, not all particles can resonate. The heavy curve in figure 1 shows the dispersion relation for the electromagnetic ion-cyclotron wave propagating along B0 in a cold electron–proton plasma(5.2)where VAp is the Alfvén speed based on the proton density (we have taken me=0 for these low-frequency waves) and Ωp is proton gyro-frequency; figure 1 is drawn for the frame moving with the bulk proton flow. In figure 1, the resonance condition (5.1) is a straight line with slope V∥/VAp and an ordinate intercept at ω/Ωp. Near the top of the figure we show the resonance conditions for two test protons, one with V∥>0 and moving with the wave, and the other with V∥<0 and moving against the wave. The intersection of the resonance condition with the dispersion relation specifies ω and k of the wave with which the particle is resonating. For a proton with V∥>0, there is no intersection and, thus, no resonance. Only protons with V∥<0 can resonate; if the waves propagate outward from the Sun, only sunward-going protons (in the bulk proton frame) will be resonant. The situation is different for heavy ions. Also shown in figure 1 are resonance conditions for three O+5 test particles, one with V∥<0 and two with V∥>0. The two lower lines intersect the dispersion relation, showing that resonance is possible for oxygen ions moving both with and against the wave. In the matter of resonant heating and acceleration, this gives the ions a major advantage over the protons, which we will discuss further below. However, figure 1 also shows that O+5 can drop out of resonance if V∥ becomes too large. As will be seen below, this presents a major difficulty.
Actually, He++ is sufficiently abundant to modify the dispersion relation. As discussed by Hollweg (2000a) and HI, He++ suffers a disadvantage similar to that of the protons, viz. only He++ particles with V∥<0 can be strongly resonant. UVCS gives no information about He++ and we will ignore this issue here.
Figure 2 illustrates several effects associated with velocity space diffusion. To keep things simple, we will here ignore dispersion. In that case, we can define a wave frame. Since there is no wave electric field in that frame, particle diffusion will conserve energy and particles will diffuse along circular arcs V∥2+V⊥2= const. (V⊥ represents the two velocity components perpendicular to B0.) In figure 2, we use the bulk proton frame, and the circular arcs are then centred on the phase speed in that frame. In reality, protons will resonate with slower-moving waves than a heavy ion such as O+5, so we have drawn separate circles for those particles. From in situ measurements of the distribution functions, Marsch & Tu (2001) have actually found evidence that solar wind protons diffuse along circular arcs; they write ‘direct observational evidence from Helios plasma data is shown for the occurrence of this pitch-angle diffusion of solar wind protons, induced by resonance with parallel ion-cyclotron waves propagating away from the Sun…parts of the isodensity contours in velocity space are well outlined by a sequence of segments of circles centred at the adapted wave phase speed…’
We will consider what happens to a group of particles which start out on the abscissa with V⊥ =0. Those particles will diffuse upward along their appropriate arcs. Upward diffusion gives V⊥≠0, and the particles acquire a perpendicular temperature; this is the most important part of the resonant heating, since perpendicular heated particles can subsequently be accelerated via the magnetic mirror force in the corona. Particles which start with the same V∥ will acquire a spread in V∥, and, thus, an increased parallel temperature, as they diffuse. (However, depending on the initial distribution, it is also possible for T∥ to decrease (Dusenbery & Hollweg 1981; Li et al. 1999). Since T∥ heating or cooling is not a major effect, we will not consider it further.) Note too that as particles diffuse upward along their arcs, they also move to the right. This represents a bulk resonant acceleration of the particles, but it is generally not as important as the magnetic mirroring once the particles acquire large V⊥. Now note the thick portions of the arcs in figure 2. They indicate the values of V∥ for which particles can be in resonance, as discussed in connection with figure 1. Only protons with V∥<0 are in resonance, while O+5 ions drop out of resonance only when V∥ is sufficiently positive. Diffusion will tend to fill up the dark arcs, but not beyond. This fact, along with the two circles having different radii, clearly allow O+5 to attain larger values of V⊥ than the protons. Thus, diffusion tends to give Toxygen,⊥/Tp⊥>moxygen/mp, in qualitative agreement with the UVCS/SOHO results (m denotes particle mass). Finally, the oxygen advantage is furthered by the fact that O+5 resonates with lower frequency waves than do the protons; lower frequency waves are expected to have more power, at least in a turbulent cascade (see §6).
These considerations are qualitatively in accord with the UVCS/SOHO data, but there are quantitative difficulties. Kohl et al. (1998) show that the O+5 temperature is an increasing function of r in 2<r/rS<3.5 (the outer limit of the data); we are assuming that the spectral line widths are due mainly to temperature. But models (Hollweg 2000a; HI) involving resonances with outward-propagating waves have been unable to reproduce this result. The resonant heating is effective in producing very hot O+5 rather close to the Sun. But the large values of V⊥ then lead to a rapid acceleration via the mirror force, causing the oxygen to drop out of resonance, as was indicated in figure 1, and to experience strong adiabatic cooling.
Resonances with sunward-propagating waves can save the day. An outward-moving ion resonating with a sunward-propagating wave will have a resonance condition represented by a straight line with negative slope in figure 1, e.g. the lowest line in the figure. As the particle accelerates outward, the slope of the straight line will become even more negative. Not only will the particle never drop out of resonance, but it will resonate with waves having smaller wavenumbers where there is presumably more power.
Another effect associated with sunward-propagating waves has been suggested by Isenberg (2001a). In that case there will be additional circular arcs in figure 2, centred on negative values of V∥. Figure 3 shows a sample of these arcs; the arcs are terminated where the particles drop out of resonance. As noted by Isenberg, the O+5 arcs overlap, while the proton arcs do not. Thus, O+5 can simultaneously resonate with sunward and anti-sunward waves, while the protons cannot. In terms of diffusion, protons diffuse only along the arcs, while O+5 can diffuse across the arcs, as indicated by the jagged dark line. This cross-arc transport is more commonly called second-order Fermi acceleration (e.g. Terasawa 1989). Since it is available to O+5 but not to the protons, it represents an energization mechanism, which is inherently preferential to heavy ions. (He++ would tend to behave like protons in this scenario.) Quantitative evaluation of the effects of sunward-propagating waves remains to be carried through.
Another puzzle concerns the ion Mg+9, which has also been extensively studied by SOHO/UVCS. Even though the charge-to-mass ratio (q/m) of this ion is close to that of O+5, the two ions appear to be heated at very different rates. Whereas O+5 is heated more than mass-proportionally relative to the protons, Mg+9 attains temperatures which are roughly mass-proportional. Why is there such different behaviour for such similar values of q/m? The answer is not known, and a full discussion of the possibilities is beyond the scope of this review; see HI and Cranmer (2002) for discussions of this issue.
6. Whence the ion-cyclotron waves?
Many researchers believe that the UVCS/SOHO data are virtually a ‘smoking gun’ for the cyclotron resonant interaction. But there is no agreement on where the ion-cyclotron waves come from.
One school of thought follows Hollweg (1986) and HJ: the Sun launches waves well below the resonant frequencies, but the resonant waves are produced via a turbulent cascade. This viewpoint is well motivated observationally. In situ observations of magnetic field and velocity fluctuations show most power at low frequencies, but with power law power spectra extending over many decades to high frequencies (e.g. Marsch & Tu 1990). The power law indices have a preference for the −5/3 value expected for a Kolmogorov turbulent cascade. In situ data for the corona are not available. However, radio studies do give some information about density fluctuations, though their connection with the putative ion-cyclotron waves is unclear. The data are suggestive nonetheless. As presented by Coles & Harmon (1989), most power resides at low frequencies (suggesting a connection with the low-frequency Alfvén waves found far from the Sun), and the power spectra have the −5/3 Kolmogorov index at higher frequencies (with some flattening as the dissipation range is approached). In both the radio and in situ data, the power spectra become much steeper at the spatial scales expected if the ion-cyclotron resonance is coming into play (e.g. Leamon et al. 1998a,b, 1999, 2000; Yamauchi et al. 1998); the spectral steepening is identified with the dissipation range in standard turbulence theory.
It is often suggested that the turbulence scenario has a serious flaw: MHD turbulence tends to produce large cross-field wavenumbers, k⊥, rather than the large values of k∥ needed for cyclotron resonance (e.g. Shebalin et al. 1983; Ng & Bhattacharjee 1996; Milano et al. 2001; Oughton et al. 2004). But theory notwithstanding, the observational fact is that there is substantial power at large k∥'s. This is especially true in the high-speed wind at low frequencies (Dasso et al. submitted), but also near the dissipation range (Leamon et al. 1998a,b, 1999, 2000). Moreover, Vasquez et al. (2004) have proposed that large k∥'s can be produced by turbulence if the background is spatially structured. Specifically, they consider an initial state consisting of Alfvén waves propagating in a background which has an inhomogeneous magnetic field and plasma, but which is in total (magnetic plus thermal) pressure equilibrium. As the wave advects the background structure, it finds itself subject to increasing refraction, and large k∥'s develop. (See also Cranmer & van Ballegooijen (2003) for a discussion of the k∥ problem.)
If the turbulence scenario is correct, the Sun must launch sufficient power at long periods to drive the fast wind; an energy flux density of 5×105 erg cm−2 s−1 at the coronal base is a representative number. This issue has been addressed by looking at Faraday rotation fluctuations impressed on a radio signal as it traverses the corona when the source is near superior conjunction with the Sun. Hollweg et al. (1982) used the linearly polarized signal from the Helios spacecraft. They were able to show that the Faraday rotation fluctuations were due mainly to fluctuations of the line-of-sight component of the magnetic field, with plasma density fluctuations playing only a minor role. They found that the polarization direction varied with time-scales of the order of hours, suggesting a connection with the long-period Alfvén waves observed by spacecraft in interplanetary space. They showed that the observed Faraday rotation fluctuations in 2rS<r<15rS closely matched what would be expected if there were indeed long-period Alfvén waves in the corona with enough power to drive the fast solar wind. Similar results were obtained by Andreev et al. (1997). However, Mancuso & Spangler (1999) and Spangler (2002) looked at Faraday rotation fluctuations from natural radio sources and concluded that there was not enough long-period wave power to drive the fast wind. But it should be pointed out that there are two basic problems with the Faraday rotation studies. The first is that the observed r.m.s. fluctuations depend on the line-of-sight correlation length for the turbulence. This length is not observationally determined and has to be guessed. The second difficulty is that the studies so far have extrapolated the data back to the coronal base without allowing for dissipation of the waves/turbulence. (For example, HI found that the wave action falls off approximately as r−1 in a simple model with turbulent dissipation.) Allowing for dissipation would lead to a larger value of the energy flux density at the coronal base.
Like the in situ data, the Faraday rotation fluctuations have time-scales of the order of hours. Similar time-scales are found for the coronal density fluctuations inferred from other radio studies. But what on the Sun is responsible for these time-scales? The 5 min oscillations and the granules have time-scales which are too short, while the supergranule time-scales are too long. We know of only one solar phenomenon which occurs on time-scales of hours: the flux cancellation events (Livi et al. 1985; Martin et al. 1985; Wang et al. 1988). It is not unreasonable to suppose that the drastic alteration of magnetic field in a flux cancellation event could launch long-period Alfvén waves with substantial energy fluxes (Hollweg 1990). This conclusion is probably closely related to a recent study by Close et al. (2004). They found that ‘the time-scale for magnetic flux to be remapped in the quiet-Sun corona is, surprisingly, only 1.4 h…implying that the quiet-Sun corona is far more dynamic than previously thought.’ In view of the long-known presence of hour periods in the low corona and solar wind, this result is not so surprising. (Hollweg (1990) was the first to suggest a connection between flux cancellation and the other observations of hour periods.)
A variation of the turbulence scenario is suggested by results of Ulrich (1996). He looked at velocity and magnetic field fluctuations in the chromosphere, and found that they were correlated consistent with outgoing Alfvén waves. The time-averaged upward Poynting flux was of order 3×107 erg cm−2 s−1 in regions of strong magnetic field. Such an energy flux is in fact about the amount required to power the chromosphere and corona in strong field active regions. The study has not been done, but it is not unreasonable to suppose that the Alfvénic energy flux in regions of weaker field, e.g. in coronal holes, might be sufficient to drive the high-speed solar wind. However, unlike the in situ data and the Faraday rotation data, Ulrich's fluctuations were mainly in the 5 min band. We are then led to ask: does the putative turbulent cascade originate from waves with periods of the order of 5 min, rather than hours?
We now turn to a completely different proposal for the origin of the ion-cyclotron waves: magnetic reconnection events in the chromosphere launch waves with the kilohertz frequencies which are resonant with protons and ions in the corona, and with the energy flux required to heat the corona and drive the solar wind (e.g. Schwartz et al. 1981; Axford & McKenzie 1992, 1996; McKenzie et al. 1995, 1997; Czechowski et al. 1998; Ruzmaikin & Berger 1998). Tu & Marsch (1997) and Marsch & Tu (1997) presented detailed solar wind models based on this direct-launching scenario. A wave with a given frequency is assumed to be dissipated at the location, where its frequency is some assumed fraction of the local proton-cyclotron frequency (see HI for a discussion of some inconsistencies with this assumption).
Tu & Marsch assume a specified magnetic power spectrum, tailored to give a spatial distribution of coronal heating which is consistent with coronal observations. They further assumed that the waves propagate along the background magnetic field. Hollweg (2000b) suggested that oblique propagation is far more likely, in which case the waves would be weakly compressive. Using a magnetic power spectrum specified by Tu & Marsch, Hollweg calculated the expected power spectrum for density fluctuations, if the waves propagate at 60° with respect to the background field. He compared this predicted power spectrum with the observed density spectrum at r=5rS given by Coles & Harmon (1989) (with a ‘correction’ so that the radio data, which were not for coronal holes, could be matched to the models, which were for holes). At high wavenumbers, the predicted density spectrum was 2–3 orders of magnitude larger than the observations. Unless the waves really are nearly parallel-propagating, which is difficult to imagine in a structured corona, we conclude that the direct-launching scenario is not viable. (See HI for a discussion of other objections to direct launching.)
A third proposal for the origin of the ion-cyclotron waves has emerged in recent years, viz. the waves are locally generated by plasma micro-instabilities in the corona; the unstable waves usually propagate highly oblique to the background magnetic field. Markovskii (2001) and Markovskii & Hollweg (2002a) considered large-amplitude long-period waves and showed that their cross-field current could drive instabilities which would be cyclotron resonant with protons and heavy ions. Voitenko & Goossens (2002a) considered instabilities driven by neutralized proton beams launched by magnetic reconnection events (i.e. microflares or nanoflares) near the coronal base. (See also Voitenko & Goossens (2002b, 2004) for some interesting work on ion heating by large-amplitude kinetic Alfvén waves. And see Viñas et al. (2000) for a discussion of electron heating by waves which are driven unstable by low-frequency waves.)
An example of this type of study (one which we believe to be particularly promising) is the work of Markovskii & Hollweg (2002b, 2004a,b). They assume that reconnection events (i.e. microflares or nanoflares) near the coronal base intermittently launch bursts of large electron heat flux outward into the corona. The heat flux drives electrostatic or electromagnetic ion-cyclotron waves unstable. These waves are highly oblique to the magnetic field, but with sunward ω/k∥ (in the local plasma frame). The cyclotron waves are strongly dispersive (with the real part of ω nearly independent of k∥), and the protons diffuse nearly along hyperbolae (not nearly circular arcs as in figure 2), giving perpendicular heating.
7. A few promising recent ideas
In §4, we mentioned that the models of HJ, Hollweg (1986), Isenberg (1990) and Li et al. (1999) were internally inconsistent in that they assumed purely outward-propagating Alfvén waves in the short-wavelength limit. In that case, the velocity and magnetic field fluctuations are related by , and the nonlinear terms which lead to turbulence, viz. , cancel. For turbulence to develop, there must be a mix of inward- and outward-propagating waves, or the waves must be long wavelength, or both. Dmitruk et al. (2002) propose a simple phenomenological model based on the following equations:(7.1)where are Elsässer variables representing waves propagating in both directions (note that convection by the solar wind flow is ignored here, along with conditions at Alfvén critical point). The term R2 (which we do not write in detail) describes how amplitudes evolve in response to the background inhomogeneity; this term is present even in the short-wavelength limit. The term R1 describes the reflections. (Except for the very last term, equation (7.1) is equivalent to Heinemann & Olbert (1980).) The last term represents the aforementioned terms leading to turbulence (except that only cross-field gradients are important). This term is approximated as (L⊥ is a transverse correlation length) giving a dissipation rate dimensionally similar to the Kolmogorov rate, but with the important difference that there is no dissipation unless both inward- and outward-propagating waves are present. Dmitruk et al. offer some numerical calculations of the volumetric heating rate in a model corona, and even derive an analytical approximation in the limit L⊥→0 (in which case the heating is proportional to ). Surprisingly, their heating rates are comparable to those found by HJ. (Dmitruk et al. also give some numerical solutions of equation (7.1) without using the analytical form of the dissipation term. See also Matthaeus et al. (1999), Dmitruk et al. (2001), Dmitruk & Matthaeus (2003), Cranmer & van Ballegooijen (2005), and Cranmer (2004).) We believe that equation (7.1) contains a significant amount of the physics needed for modelling, while at the same time offering the virtue of simplicity (an extension to include the solar wind flow and conditions at the Alfvén critical point has been made by Cranmer & van Ballegooijen (2005)).
An approach such as Dmitruk et al. (2002), though valuable, still does not address kinetic aspects of the heating, such as were outlined in §5. The goal would be to calculate how the proton and ion distribution functions evolve subject to resonant diffusion in velocity space, along with global forces such as gravity, magnetic mirroring, etc. Such a calculation would also entail knowing the evolution of the wave power spectrum, especially in the resonant dissipation range. No such calculation is available. However, Isenberg et al. (2001) and Isenberg (2001b, 2004a,b) have suggested an approximate procedure, which is mostly analytical. They call their approach ‘the kinetic shell model’, the shells being the three-dimensional versions of the diffusion arcs in figure 2.
The essence of this model is shown in figure 4, which is similar to figures 2 and 3. Velocity space (V∥, V⊥) is displayed, along with several diffusion arcs for protons. The kinetic shell model's key assumption is that the velocity space diffusion is faster than any other time-scale, so that the distribution function is always very nearly uniform along each arc (or shell). (This implicitly assumes that the wave power spectrum in the resonant range is above some minimum level.) However, the number of particles can differ from shell to shell. Individual shells can be thought of as moving in response to the global forces: gravity, mirroring and the charge-separation electric field; the sum of gravity and the electric field is always sunward.
Consider first sunward-moving protons, which we take to have V∥<0. Recall that these protons resonate with anti-sunward propagating waves. The leftmost shells, which reach to large values of V⊥, experience a net mirror force which overwhelms gravity and the electric field, so those shells are pushed to larger values of V∥, as indicated by the leftmost arrow. Other shells having V∥<0, but reaching to smaller values of V⊥, are dominated by the sunward combination of gravity and electric field; those shells are driven to more negative values of V∥, as indicated by the second arrow from the left. The result is that the sunward-moving protons tend to accumulate near a shell where all forces balance, the ‘pile-up surface’.
At V∥=0, the protons are resonant with waves having infinite wavenumber (figure 1) and zero power. Thus, those protons are unaffected by the waves, and move across the V∥=0 ‘boundary’ in response to the global forces. If V⊥ is large enough, the particles will be pushed toward V∥>0, as indicated by the right-pointing arrow; conversely for particles with lower V⊥.
Particles with V∥>0 resonate with sunward-propagating waves, and follow the diffusion arcs sketched in the figure. The two rightmost arrows indicate the shell motions in response to the global forces; shells which reach to large V⊥ are pushed to larger values of V∥, and so on. In this case, the opposite of a pile-up surface develops: there will be a ‘rarefaction surface’ on which particles are depleted.
Even if there are no sunward-propagating waves initially, in this model they will be generated by those protons at large V⊥ which are pushed into the V∥>0 region by the mirror force. The resulting distribution is highly unstable to the generation of sunward-propagating waves; the protons appearing at V∥>0 will diffuse down and to the right in the figure, losing energy which is made available to the sunward waves.
One virtue of the kinetic shell model is that it implicitly contains information about the waves, which does not need to be put in ad hoc. Since the resonant interaction conserves the energy of the combined wave–particle system, the evolving particle distribution can be used to determine the dissipation rate, or generation rate, of the interacting waves.
Initial results (Isenberg 2001b) were very promising. With dispersion ignored, and the waves propagating at VA, rapidly accelerating fast winds could be produced. But when wave dispersion was included (Isenberg 2004a,b), the results changed qualitatively: only slow winds could be obtained. The reason has to do with the tops of the arcs in figure 4. Those particles have large V⊥ and experience a strong mirror acceleration. But those particles also have small V∥. According to figure 1, they resonate with waves that have slow phase speeds. Thus, according to figure 2, their diffusive arcs have small radii. The net result is that the tops of the arcs in figure 4 flatten out when dispersion is taken into account. The diffusive arcs then extend to smaller values of V⊥ than in the dispersionless case, and the net mirror force is reduced.
Nonetheless, in the author's opinion, the kinetic shell model has many virtues. Perhaps further developments including the warm plasma dispersion relation and obliquely propagating waves will save the day.
We have shown how our thinking about the solar wind has progressed from Parker's wind driven by the electron pressure gradient, through a wave-driven wind driven mainly by the pressure of Alfvén waves, to our current idea that the rapid acceleration of the fast wind in coronal holes is due mainly to the pressure tensor of hot protons, which is mainly the magnetic mirror force if the protons are highly anisotropic with T⊥≫T∥ (see Vásquez et al. 2003 for a discussion of the importance of thermal anisotropy). Our current picture also allows for a significant contribution from wave pressure, but this will give a more gradual acceleration acting mainly beyond the sonic point.
We have emphasized that the wind is undergoing significant heating in the region where it is rapidly accelerating to supersonic speeds. Coronal heating and solar wind acceleration are intimately linked and need to be treated together.
The UVCS/SOHO fast solar wind data have shown that coronal heating works mainly on the transverse (to B0) components of protons and ions. It is not Joule heating, as is very commonly presumed to be the case in other parts of the corona, such as the active region loops. Coronal heating also does not seem to be dominated by viscosity or by heat conduction. We are certainly led to ask whether proton and ion heating dominates other parts of the corona, such as the active region loops, as well. (There is already some evidence that the same processes are at work on the open field lines in coronal streamers, but not on the closed field lines; see Frazin et al. (2003).)
Heating transverse to the magnetic field strongly implicates the ion-cyclotron resonance. But we still do not know the source of the high-frequency resonant waves. We have given some arguments against the proposal that the Sun directly launches these waves, with kilohertz frequencies, but the jury is still out. The Sun could well launch the required energy fluxes at long periods (minutes to hours), with the high frequencies generated by a turbulent cascade. The tendency of MHD turbulence to produce mainly high k⊥ is a difficulty, but we do not believe it is insurmountable. Finally, a number of recent papers have explored the promising idea that the high-frequency waves may originate locally in the corona from plasma micro-instabilities; the instabilities may themselves be a consequence of distortions of the proton or electron distribution functions which result in response to reconnection events at the coronal base. The self-consistent calculation of the evolution of the wave spectra and particle distribution functions is a formidable but necessary task; preliminary studies show no contradictions with observations, so the instability scenario seems worth pursuing.
One thing should be clear from the foregoing discussion. A complete description of coronal hole heating and fast wind acceleration will require detailed considerations of particle kinetics; MHD does not tell the whole story. The non-Maxwellian particle distribution functions observed by spacecraft (e.g. Marsch 1991) are trying to tell us something about the physics of heating and acceleration closer to the Sun, but their message has not yet been decoded.
Future studies will need to place more emphasis on wave propagation oblique to the magnetic field. The Sun almost certainly launches highly oblique waves, which are subsequently refracted. And the micro-instabilities considered so far tend to produce highly oblique waves.
Future studies will also need to expand consideration of inward- as well as outward-propagating waves. The micro-instabilities can produce inward-propagating waves. Outward-propagating waves launched by the Sun are expected to have long wavelengths, and so they will be easily reflected. The mix of inward and outward waves is an essential part of the development of turbulence; similarly for the second-order Fermi process believed to be important for the heavy ions.
As a final remark, we need to say something about electron heating, which has been completely ignored in the foregoing. It is difficult to say much about the electrons because we do not understand how to model their heat conduction in the weakly collisional conditions found in the corona and solar wind. Consequently, we do not know how to translate observations of their temperature variations in the corona into heating requirements. It is possible that the electrons need no external heating, and that collisional coupling with the protons coupled with efficient heat conduction suffices to maintain their temperatures, which are observed to be somewhat below 106 K close to the Sun in coronal holes (Wilhelm et al. 1998). It is also possible that the electrons are strongly heated intermittently by reconnection events at the coronal base, as in the micro-instability scenario discussed above (Markovskii & Hollweg 2002b, 2004a,b). And it may be that electrons are heated throughout the corona via kinetic Alfvén waves. These are basically Alfvén waves which propagate highly obliquely to the ambient magnetic field, but which are compressive (Hollweg 1999) resulting in a fluctuating parallel electric field. Since the electron thermal speed is comparable to VA in the corona, the electrons can be heated via the Landau resonance (e.g. Leamon et al. 1999, 2000).
An entirely different approach has been taken by Fisk (2003) and Gloeckler et al. (2003). They emphasize observational evidence that the fastest (slowest) solar wind originates from the coronal regions with the coolest (hottest) electrons, a result completely opposite to the original model of Parker (1958). They explain this result in terms of ‘interchange reconnections’ which convert closed loop-like magnetic field lines into open field lines. Plasma originally on the loops is, thereby, made available to the solar wind. Space does not permit us to reproduce their arguments, but the net result is that the hotter loops contain more mass than the cooler loops, and with more mass to accelerate, slower flows result. Alternatively, Schwadron & McComas (2003) have suggested that coronal regions with hotter electrons give rise to slower solar wind, because those regions conduct more energy down to the transition region where it is lost via radiation, leaving less energy for the kinetic energy of the flow. Finally, the electron temperature–speed anticorrelation might be consistent with the idea that it is mainly the protons which are heated (by the mechanisms discussed in this paper) in all regions which give rise to the solar wind. The protons then heat the electrons via collisions. Less dense regions, viz. the coronal holes, have weaker collisional coupling between electrons and protons, and this presumably results in lower electron temperatures coupled with faster flows.
The author is grateful to S. R. Cranmer, P. A. Isenberg, M. A. Lee, S. A. Markovskii, W.H. Matthaeus and B. J. Vasquez for many helpful conversations and collaborations. This work was supported by the NASA Sun–Earth Connection Theory Program under grant NAG5-11797 to the University of New Hampshire.
One contribution of 20 to a Discussion Meeting Issue ‘MHD waves and oscillations in the solar plasma’.
- © 2005 The Royal Society