Computer Physics Reports 12 (1990) 205-232 North-Holland

205

HEATING OF THE SOLAR CORONA

Joseph V. HOLLWEG

Space Science Center, Institute for the Study of Earth, Oceans and Space, University of New Hampshire, Durham, NH 03824, USA

We review a number of models which are currently being considered for coronal heating, but we consider also heating of the chromosphere which requires nearly as much energy as the active corona, and more energy than

coronal holes or the quiet corona. There are basically two types of models, which are motivated by a variety of observations. (1) Models which invoke MHD waves generated by the convective motions are motivated by observations of the ubiquitous presence of Alfven waves in the solar wind. There is evidence that these waves heat and accelerate the solar wind protons and heavy ions. The solar wind thus provides one example of wave heating. Waves have the advantage of being able to heat the chromosphere and photospheric magnetic flux tubes on their way to the corona. MHD turbulence (as observed in the solar wind) or resonance absorption seem to provide adequate dissipation mechanisms. A problem with wave theories is that the waves tend to be reflected by the steep AlfvCn speed gradient in the chromosphere and transition region, but it is estimated that adequate energy fluxes can enter the open corona, or closed coronal loops if global loop resonances can be excited. Short coronal loops (L I< lo4 km) can also receive adequate wave energy fluxes even if the loop resonances are not excited, but a problem exists with getting enough energy into intermediate length loops (L - 104-5 x lo4 km) since their resonant frequencies are possibly to high to be excited. (2) Models which invoke the gradual buildup of coronal magnetic energy due to random walks of the photospheric flux tubes, and the subsequent release of that energy via current sheet formation and reconnection, are supported by observations indicating that localized impulsive heating and dynamic events occur in the transition region and corona. These models cannot explain the chromospheric heating or the coronal heating on open field lines. They require substantial random walks of the photospheric footpoints, which still need to be observationally verified. A third possibility, which has not been studied in detail, is that the chromospheric and coronal heating is associated with emergence and cancellation of magnetic flux. All types of models are ripe for further studies using numerical simulations, and along the way we shall offer several suggestions for fruitful numerical studies.

0167-7977/90/$9.80 0 1990 - Elsevier Science Publishers B.V. (North-Holland)

206

Contents

1. Introduction. . . . , . _ . . .

2. Observational constraints 3. General considerations .

4. Magnetic stresses . . , . . .

5. Waves . . . . . . . . . . . . . .

6. Conclusions . . . . . . _ . . .

References _ _ . . . . . . . . . . .

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J. V. Hollweg / Heating of the solar corona 207

1. In~~uction

It has been some 50 years since Grotrian and EdlCn showed spectroscopically that the solar corona contains some very highly ionized heavy elements, thus demonstrating that the coronal temperature is in excess of 10” K. Yet, in spite of a half century of study, it is still not known how the megakelvin temperatures of the corona are maintained against losses due to electron heat conduction, radiation, and energy outflow into the solar wind. Moreover, we shall see below that the coronal heating problem has been inappropriately emphasized in the past, because the energy requirements of the chromosphere and high-speed solar wind are nearly as large as the coronal energy requirements. From an energetic standpoint at least, the corona should not be viewed in isolation, and it is probably necessary to consider the chromosphere, corona, and solar wind together. It is thus even more disappointing to realize that the chromospheric heating is also not understood, and that we still do not know the energy source of the solar wind (in particular the high-speed solar wind streams).

However, since the Skylab observations of the early 1970s it has become apparent that the upper chromosphere, corona, and solar wind do not derive their energies from acoustic waves and their attendant shocks, as was long thought to be the case. This at least is progress (although there has been some recent work arguing that much of the chromosphere may be heated by acoustic waves after all [1,2]!).

This has focused attention on the role of the magnetic field in heating the solar atmosphere. It has also been known for some 20 years that the solar magnetic field is highly structured, with most of the magnetic flux in the photosphere being clumped into small intense flux tubes, with magnetic field strengths probably in excess of 1000 G and diameters of the order of 200-300 km. And even in regions having a dominant magnetic polarity, one can still find the ubiquitous presence of flux tubes with the opposite polarity, which appear to interact dynamically with tubes having the dominant polarity. All of this has focused attention in recent years on the roles of inhomogeneity and complicated magnetic topology in heating the solar atmosphere.

Solar wind studies too bear on the issue of coronal heating, since the solar wind is the only part of the corona which can be directly sampled, and it is the only place where the all-important vector magnetic field can be measured with high time resolution. For 20 years it has been known that the wind contains large-amplitude AlfvCn waves, which are probably of solar origin. This has provided strong impetus to models which invoke magnetohydrodynamic (MHD) waves to heat the solar atmosphere. But work in the past decade has emphasized the remarkable observation that the magnetic field power spectrum in the solar wind is a power law over many decades of frequency (or wavenumber). This has been interpreted as the inertial range in a turbulent energy cascade, and it thus appears that the solar wind irregularities have a curiously dual nature consisting of waves and turbulence. This suggests that MHD turbulence may also play a role in heating the solar atmosphere.

Magnetic fields, inhomogeneity, and turbulence, have thus been recurring themes in recent attempts to explain the heating of the solar atmosphere. Many new ideas have been advanced, and it is an exciting time for the solar theorist. In the following pages we will attempt to summarize some of the current concepts which are being considered. However, we shall emphasize that there is still no “smoking gun” supporting any of the theories which have been advanced. Part of the problem is that virtually every theory involves dynamics at spatial scales

which are too small to be observed, and a time-varying magnetic field which is difficult to observe anywhere, and there is thus rather little definitive data to guide the theorist or test a model. But all theoretical considerations are implicating complicated geometries and nonlinear- ity, and it may be that real progress can only be made via numerical simulations: we shall try to emphasize areas where simulations may be particularly helpful.

The overall plan of this review is to first summarize the basic observational constraints. We shall then review the salient features which everyone agrees should be part of theoretical models of solar atmospheric heating. We will then summarize areas where the theoreticians have taken different approaches, the reasons for the differences, and the successes and failures of the various ideas which have been advanced. We shall see that there are basically two approaches, viz. models which invoke MHD waves, and models which invoke quasi-static deformations of the magnetic field leading to current sheet formation and heating by reconnection.

The goal of this review is to provide a survey of current ideas rather than a complete survey of the literature. We will, however, cite papers which can serve as further guides to the literature. The reader should also consult other recent reviews on solar atmospheric heating [3--61.

2. Observational constraints

Consider first the coronal magnetic field. Its overall morphology is revealed by photographs taken during solar eclipses. The brightest coronal regions (with the highest electron densities) are the helmet streamers, which basically outline closed magnetic field, i.e. loop-like magnetic field lines which return to the solar surface after reaching heights of no more than about one solar radius. (These regions are also called guier coronal regions.) However. their outermost field lines appear to be open, extending more-or-less radially outward into the solar wind. These open field lines also contain a relatively high electron density, and they give these regions a spiked appearance, reminiscent of World War I Prussian helmets. From solar wind studies, it has been determined that the spikes are essentially current sheets, across which the interplanetary magnetic field changes polarity; they are associated with the slower, denser solar wind flows. Also apparent on eclipse photographs are darker regions (with lower electron densities) which appear to be associated with open magnetic field lines which extend outward into the solar wind. (The electron densities are presumably lower because there is a continual loss of plasma into the wind.) These regions are called coronal holes, and they are known to be the sources of the high-speed solar wind streams.

In X-ray coronal images, the coronal holes are even more apparent, because the X-ray emission is proportional to the square of the electron density. The X-ray images also show that the densest coronal regions are small loops, with typical lengths of the order of 10’ km or less. and typical diameters of the order of one tenth their lengths. The loops obviously outline closed magnetic field lines, and they tend to be associated with solar active regions in the general vicinity of sunspots; they are thus called active region loops. However. it is important to realize that the active region loops are really loops of dense plasma embedded in a larger scale magnetic field of roughly uniform strength: this is necessarily the case since the loop plasma pressure is much less than the estimated magnetic pressure. Also apparent on the X-ray images are much smaller regions of intense emission called X-r+) brig& points. They appear to be very small

J. V. Hollweg / Heating of the solar corona 209

loops, or assemblages of loops, which are shorter lived than the larger active region loops and which exhibit temporal variability on time scales of minutes [7]. The bright points can appear even in the coronal holes, and they then appear to be associated with coronal plumes which are rays of denser plasma extending outward into the solar wind.

The active region loops have the largest heating requirements, and they pose the greatest challenge for the theorist. Their association with the stronger active region magnetic fields (50-100 G) is one of the reasons for believing that the magnetic field is essential. And the loop-like morphology of the X-ray corona is one of the principle reasons for regarding structure as an important ingredient in the heating process.

Consider next the photospheric and chromospheric magnetic field. The photospheric field is clumped into fhx tubes with fields estimated to be in excess of 1000 G. The magnetic pressures inside the flux tubes are comparable to the internal plasma pressures, which are thus about half the external plasma pressure. The field lines in the photosphere are essentially vertical, because they are buoyed up due to the lower plasma density in the flux tube. Since the plasma pressure decreases rapidly with height in the photosphere and chromosphere, the magnetic field is less effectively confined and the flux tube fans out. At a height of only about 500 km above the photosphere, the field lines near the edge of the flux tube have become essentially horizontal; this is called the magnetic canopy. They then turn approximately vertical again after they interact with the expanded field lines from neighboring flux tubes. In active regions the average field strength is 50-100 G while in coronal holes the average field strength is 5-10 G.

The flux tubes are concentrated along the edge of the supergranular convective cells. This is also where one finds excess photospheric and chromospheric heating, and the solar spicules. This association is another reason for believing that the magnetic field is responsible for heating and for dynamic phenomena such as spicules. Cook and Ewing [8] have recently shown that excess heating (in erg cmP2 s-i) near the temperature minimum region (at a height of about 400 km) is linearly correlated with the magnetic field strength in the underlying photosphere; they suggest that the linear relationship is consistent with heating by AlfvCn waves, since their group velocity is proportional to B.

Any theory of heating of the solar atmosphere must account for the following energy flux densities, in erg cme2 s-i [9,10]:

quiet corona: = 3 x 105, coronal hole with high-speed wind: = 8 x 105, chromosphere = few X 106, spicules: = few X 106, active region loops: = 10’. Note that the chromosphere, spicules, and high-speed solar wind require nearly as much

energy as the active region loops. Energetically, the coronal heating problem should not be isolated. Note too that the chromosphere is only some 1500 km thick, so the average volumetric heating rate is about 3 X lop2 erg cmP3 s-l, compared to about 10e3 erg crne3 SK’ in active region loops.

Theories which invoke waves are constrained by the observed nonthermal motions derived from the widths of spectral lines. In the corona these tend to be in the range 20-40 km ss’, rms, along the line-of-sight [11,12]. This critical piece of information is very sketchy. It would be very useful to know how these velocities are correlated with individual loop heating rates and with

magnetic field strength. Nonetheless, it can now be said that these motions cannot supply the above energy requirements if they were associated with sound waves. Again, we must appeal to the magnetic field.

Further observational constraints become even vaguer. There is general agreement that heating is positively correlated with field strength. Cook and Ewing have found this to be the case in the temperature minimum region. Golub et al. [13] have found that the active region loop pressures scale as B'.'. which is roughly consistent with a model in which the coronal volumetric heating rate scales as B2, but their data set was very scanty and more definitive data is needed. But this is unfortunately one of the great stumbling blocks of attempts to model heating of the solar atmosphere: everyone agrees that the magnetic field is vital, but the best we can do is estimate the coronal and chromospheric field strengths, and definitive extensive datasets are lacking. Nonetheless, further studies of the type done by Golub et al. are needed, perhaps augmented with measurements of the nonthermal velocities in individual loops. Of course. wave theorists would particularly like to know how the magnetic field vector .f/uctuutes in the corona, but there is little observational hope in this regard.

There is also general agreement that active region loop heating is inversely correlated with loop length. Parker [6] has remarked “one of the more astonishing features of the active corona.. . is that the surface brightness of the active regions is approximately independent of the dimensions, from the small ephemeral active region with a characteristic scale L = 10" km to the large normal active region with a characteristic scale L = 10' km or more.” Even this statement is rather general, and a quantitative statement relating heating rate to loop length is not available. It might even be that Parker’s observation reflects nothing more than an inverse correlation between B and L. However, we shall see that most current theories do predict stronger heating on shorter loops.

Everyone agrees that the motions of the magnetic field footpoints is a critical factor in heating the solar atmosphere. since the heating must ultimately come from the work done by the convective motions on the magnetic field. This obviously applies to wave theories in which waves are launched by the twisting or shaking of the photospheric flux tubes, and it also applies to theories which involve topological changes in the coronal magnetic field brought about by slow rearrangements of the photospheric flux tubes. But again we encounter a major stumbling block: the photospheric flux tubes are too small to be resolved by current instruments, and their motions are not known. We need to know whether there are twists. the amplitudes and frequency spectra of the motions, and the statistics of their random walks.

We also need better information on the field strengths in the photospheric flux tubes, and how they evolve with time. In the latter regard, an important aspect may be the fact that even regions on the sun having a dominant magnetic polarity also contain flux tubes of the opposite polarity. It has been observed [14-211 that small regions of opposite polarity can approach one another in the photosphere, and cancel, with essentially no local magnetic flux remaining after the cancellation. Magnetic energy is evidently being destroyed, probably via reconnection, and the released energy may be paramount in producing dynamic events and heating in the solar atmosphere. Small flares have been associated with these flux cancellation events. The cancelled magnetic flux can be replenished by the emergence of small bipolar regions from below the photosphere, so that a quasi-steady state can be sustained. The roles of flux emergence and cancellation have generally been ignored in theories of chromospheric and coronal heating,

J. V. ~ollweg ,/ Heating of the solar corona 211

although it may be the key missing ingredient in our theoretical concoctions. On the other hand, the author has seen video magnetograms showing large regions of strong photospheric magnetic field of one polarity, but flux cancellation was not evident. These active regions were presumably accompanied by significant chromospheric and coronal heating, and I would find it difficult to ascribe that heating to flux cancellation events. Perhaps the cancellation was unresolvable on the magnetogram. (Or perhaps the author is ill-trained to interpret video magnetograms!) Wang et al. [22] note a preference for the flux cancellation events to occur along the main magnetic neutral line in an active region.

In any event, the observed fact is that some coronal field lines give bright X-ray emission and contain lots of plasma, while surrounding field lines are darker in X-rays and contain less plasma. Why some field lines light up and others do not is not known. It must be related to the conditions at the footpoints of the field lines, but observations have not yet told us what is really happening there.

Finally, we mention a few relevant plasma parameters which the theories have to explain. The active region loops have temperatures of about 2.5 X lo6 K, and plasma pressures between 0.5 and 10 dyne cmp2 [23]. We will use a median pressure of 2 dyne cmw2, corresponding to a density of 5 X lo-l5 g cm-j (for a molecular weight of 0.5). The density outside the active region loops is lower by a factor of about l/3; the temperatures are uncertain. The coronal holes are cooler, with maximum temperatures of about 1.5 X lo6 K at about half a solar radius above the surface. The coronal hole base pressures are about 2.5-6 X 10V2 dyne cmp2 [24]. For most studies of coronal heating, the plasma can be regarded as collisional and strongly magnetized. Following Braginskii [25], the proton collision time in an electron-proton plasma is or = 0.75Tr’/2n;’ s in c.g.s. units, for a Coulomb logarithm of 22. In an active region loop we have typically rp = 1 s, the proton mean-free-path S is about 150 km, ocprr, = 5 x lo5 in a 50 G magnetic field (w,r is the proton cyclotron frequency), and the proton cyclotron radius is only 40 cm. At the base of a coronal hole ( Tp = lo6 K, nr, = 2 X lo* cme3) rp is only increased to = 4 s, and ticr,rr, = 2 X lo5 in a 6 G field. Similar considerations apply to the electrons, whose collision times are shorter by a factor of about 60-l. However, above a height of about half a solar radius in a coronal hole, collisionality breaks down and kinetic effects may be important. In particular, the classical expression for electron heat conduction becomes highly suspect [26,27]. For further information on active regions and coronal holes, see the Skylab Workshop proceedings edited by Zirker 1281 and Orrall [29]. For models of the chromosphere, see Avrett, ref. (lo]. For a recent review of chromospheric heating see Narain and Ulmschneider, ref. [5].

3. General considerations

The old scenario for coronal heating was that the varying pressure field in the convection zone generated sound waves which steepened into shocks, giving global heating (because the shocks propagate through the entire atmosphere). It is now believed that the magnetic field is the starting point, but there is no agreement on whether waves are involved. It is agreed, however, that heating must occur at small spatial scales, because the electrical resist&y and shear viscosity coefficients are very small in the chromosphere and corona. However, shocks are probably not involved, because the observed nonthermal motions are small compared to the

AlfvCn speeds in the corona, implying weak shocks with small entropy jumps. and because the shocks propagate rapidly so that their heating is spread out over very large volumes and small structures such as the active region loops can not be explained. (There is some faint hope for

shock heating in coronal holes, however: see ref. [30].) Thus the heating is now envisioned as being rather local, and in some cases it becomes necessary to devise ways to spread the heat out.

One exception to the above remarks deserves mention. In a weakly collisional magnetized plasma such as the corona there are five viscosity coefficients. denoted qo,. . . . q4 by Braginskii [25]. The largest by far is Q, and its magnitude is lO-“T’,“’ in c.g.s. units, where TP is the proton temperature in a fully ionized hydrogen plasma. The Reynolds number is R = pVD/v,,. where p, V, and D are typical values for the density, velocity, and length scale. If we take active region loop parameters TP = 2.5 X 10” K, p = 5 X 10 I5 g cm ‘, I/= 30 km s ‘. and D = lo4 km, we obtain R = 15. This surprisingly small value suggests that q. might play an ilnportant role in coronal dynamics. The problem is that q(, comes into play only for certain types of motions in which v l V# 0 or V,, # 0 (the subscript /I refers to the component along the magnetic field B) [31,32]. It is thus not effective in dissipating Alfven waves or the motions associated with resonance absorption (see section 5 below). However, there may be situations where it is important, in reconnection perhaps, and it should not be ignored, especially in numerical simulations. In fact, the author is surprised that there are so tnany studies which work very hard to simulate the effects of magnetic Reynolds numbers of lO”‘-IO’“, while completely ignoring the effects of a viscous Reynolds number which is at least 10 orders-of-magnitude smaller. It is hoped that future numerical and analytical work will pay more attention to Q. And even Reynolds numbers based on Braginskii’s v3 and r4 are much smaller than the magnetic Reynolds numbers.

Thus understanding the heating of the solar atmosphere basically boils down to answering these two key questions:

(I) How does energy get into the chromosphere and corona? (2) How does it dissipate at small spatial scales? In the next two sections we will summarize current attempts to answer these questions in

terms of magnetic heating; we will not review acoustic models of chromospheric heating [1,2.5].

4. Magnetic stresses

In the last decade, a number of models have been proposed which do not involve MHD waves. There are basically three reasons why some workers have rejected MHD waves.

First, the requirements on the dissipation rate are rather stringent. Consider an active region loop requiring 10.-‘erg cmP3 se’. We estimate the wave damping rate y by requiring

p(6V2)2y = 1o-3

where 6V is the wave velocity fluctuation. the angle brackets denote the variance, the factor 2 comes in because y refers to the amplitude while the above expression refers to energy, and we have assumed equipartition between kinetic and magnetic energies. We take p = 5 x lo-- ” g cm-3 and (6V”) = 2 x (30 km s’)‘, where the factor 2 allows for two degrees of freedom

J. V, Hollweg / Heating of ihe solar corona 213

transverse to B. We obtain y--l = 180 s. If we take B = 50 G, the Alfvitn speed u, is 2000 km s- ‘, and uA/y = 3.6 x lo5 km, i.e. the waves have to dissipate in only a few active region loop lengths. We shall see below that this is not an insurmountable difficulty.

Second, the Alfven speed increases rapidly between the photosphere and corona, and this leads to strong wave reflections and difficulties with getting enough wave energy into the corona 1331. We shall see below that in most cases there are ways out of this quandary, although some significant problems remain.

Third, for compact loops, such as the X-ray bright points, the period required for half a wavelength to fit into the loop is only 10 s, if we take u, = 2000 km s-l and L = lo4 km. This is the period of the fundamental resonance of the loop, which can act as a resonant cavity because waves are strongly reflected off the dense chromospheres at each end of the loop. It is unlikely that the convection zone launches much wave power at such short periods. At longer periods it is easier to think of field lines as simply being quasi-steadily displaced by slow footpoint motions.

Parker [34] imagined an active region loop to be straightened out, with a dense photosphere and chromosphere at each end. A field line at one end of the loop is taken to be fixed in the photosphere, while the footpoint at the other end is displaced ho~zontally at a velocity Vb_ The displaced field line is taken to remain straight. If the magnetic field has no horizontal component at t = 0, its horizontal component at later times is BoVht/L, where L is loop length and B0 is the initial field strength. The magnetic energy density increases as t2, and the energy flux density into the loop is

F = B;V&WirL. (1)

Now, what should we put in for t? In a long series of papers (see refs. [6,35,36] for references), Parker, Low [37], Wolfson [38], and others have argued that the continual braiding of flux tubes as they randomly walk through the photosphere must necessarily lead to the formation of current sheets in the corona. The current sheets will lead to reconnection, which relaxes the stresses which have been built up according to eq. (1). The reconnection eats its way through the flux tubes at a velocity u,,,, and a steady state will be reached if we take

t = D/u,,

in eq. (l), where D is the flux tube diameter. Thus

F = B;V$3/4nu,,L. (2) The appearance of L in the deno~nator is {qualitatively, at least) a desirable feature of this

and other related models. Somewhat surprising is the appearance of u,,, in the denominator. Physically this comes about because rapid reconnection means that the field easily relaxes back to a nearly vertical configuration, and the convective motions can not do much work against a nearly vertical field.

Let us take F = lo7 erg cm’ s-i for an active region loop, D/L = 0.1, L = lo5 km, B0 = 50 G, 0, = 2000 km s-r, and V, = 0.4 km s-i (a typical velocity associated with the supergranulation). EZq. (2) then requires u,,/u+, = 1.6 x lo-‘, t = 87 h, and Vht = 0.18 solar radii. The required reconnection rate is surprisingly small, but Parker f34] notes that it is well within the bounds of the maximum and ~nimum estimates for u,,,/u~. Moreover, Parker 1351 has suggested that the

reconnection might proceed in rapid but short-lived bursts, leading to small flares, and that the above value of ~,,,‘t’~ merely represents an average. Perhaps more worrisome is the large value of Y,,t. Video magnetograms do not indicate that the photospheric flux tubes shuffle and intermix to this extent [39], but it should be kept in mind that the individual photospheric flux tubes are not resolved in these data. Also, it may be that the mesogranulation, which is difficult to observe, is most effective in moving the magnetic footpoints [40].

Parker [35] has summarized recent evidence indicating that at least some energy is released quasi-impulsively into the corona. These include observations of hard X-ray spikes with dura- tions of 1-2 s. spatially localized brightenings of ultraviolet spectral lines lasting 20-100 s, and ultraviolet observations of jets and turbulent events which are of the order of lo3 km in size and which last 1-2 min. And we have already mentioned the time variability of the X-ray bright points [7], with time scales of several minutes, in both EUV and X-ray emission. The observed energies in these impulsive events generally fall far short of the required energies to heat the active region corona. However, the less-energetic hard X-ray spikes are more numerous, and most of the emitted energy occurs in the smaller spikes. down to the detection limit of about 10’” ergs per spike. Parker has called these small spikes nanoflares. and he suggests that what we see as the X-ray corona is simply the supe~osition of a very large number of nanoflares. with most of the energy being released at energies below the threshold of current detectors. Future observations with more sensitive detectors will obviously be crucial to test this idea, but the general picture is consistent with Parker’s proposal that motions of the magnetic footpoints inevitably lead to current sheets which subsequently lead to reconnection and the nanoflares.

Parker’s ideas have been challenged by several workers, who dispute Parker’s claim that current sheets are inevitable. Antiochos 141.423 has argued that current sheets will form only at magnetic null points (where B = 0) or at surfaces of discontil~uous magnetic connectivity: these surfaces separate field lines which start out in the photosphere with footpoints which are very close together, but which connect to very different regions of the photosphere at their other ends. Aly’s studies [43] concur with Antiochos. Current sheets formed in this way probably cannot explain the heating of active region loops, since these loops outline field lines with the same magnetic connectivity.

Van Ballegooijen [44] has suggested that magnetic energy cascades to small scales via a statistical process, rather than by the formation of current sheets. He starts with the same basic idea of Parker, viz. the random walks of the photospheric flux tubes lead to continual braiding of field lines in the corona. Statistically, the separation of neighboring points increases exponen- tially with time, and the power spectrum of the electric current density at large w~~~~enumbers also increases exponentially with time. Eventually the current at very small spatial scales leads to

joule heating. Van Ballegooijen’s expression for the energy flux density into the corona differs from Parker’s

(ey. (1)). Eq. (1) was derived by assuming that the footpoint motion simply changes the length of a field line. Van Ballegooijen assumes that the field lines get braided with the number of braids, IV, increasing linearly with time. If each braid consists of a horizontal field line displacement 13. then the rms horizontal magnetic field is roughly given by

J. V. Hollweg / Heating of the solar corona 215

The number of braids is N = t/t,, where t, is the braiding time, which is taken to be the time for a random walk to displace the footpoint the distance D:

t, = D2/V&, (4)

where r is the correlation time for the random walk. Eqs. (3) and (4) give the following expression for the energy flux density into the corona:

F = (V&D)‘[ @V;t/4=L]. (5)

Note that the quantity in square brackets is Parker’s expression, eq. (1). Note too that eq. (5) is very sensitive to the value of V,, which is not a well-known parameter. Van Ballegooijen suggests that observations of the diffusion of the photospheric magnetic field can be used to estimate V&,

which he takes to be 240 km* s-i. If r is taken to be a granule lifetime of about 500 s, this implies V, = 0.7 km s-’ which is not unreasonable. For D we take the diameter of an elemental flux tube after it has expanded into the corona, since the flux tubes can be regarded as the individual “ ropes” forming the braids. If the photospheric diameter is 300 km, and if the field strength declines from 1500 G in the photosphere to 50 G in the corona, then D = 1640 km, and (Q/D)* = 0.045. In van Ballegooijen’s approach, F is much less than the value used by Parker, and larger values of t and V$ are required to supply the steady-state flux of 10’ erg cmm2 s-l. (However, we expect D2 a IIt1 and thus F a Bi in eq. (5), and stronger coronal fields can significantly reduce t.) In the steady-state, the number of braids is

A’= ~ITLF/B,~V;T. (6)

If we take the above numbers and L = 10’ km, we find N = 200. Similar, but much more sophisticated considerations have been given recently by Berger [45].

To continue with van Ballegooijen’s analysis, he takes a steady-state to be reached when

t = D2(log R,)/6(2~r)“~V&

where R, is the magnetic Reynolds number, which is about 10” in the corona. The logarithm appears because the current density at small scales increases exponentially with time. We finally obtain then

F= 5.3 x 1O-3 (log R,)B,fV;q’L.

For the typical parameters we have been using, we obtain F = lo5 erg cme2 s-l, which is too small by two orders of magnitude.

Another model involving random walks has been given by Sturrock and Uchida [46], but they consider the random walk to lead to a gradual twisting of a thin axisymmetric flux tube. This implies generation of a current flowing along the tube. Let the current inside radius r be I, so that be azimuthal field is B, = 2I/rc, where c is the speed of light. We will assume that the axial

field, B,, is uniform in any cross-section of the tube. The magnetic flux inside r is @ = T~‘B,,

and thus

Since I and Qt are constants, the pitch of the twisted field lines is greatest on the expanded part of the tube. The equation for a field line is Y d8/dz = B,/B,, and after integrating over the length, L, of the flux tube we find that the field line has rotated through an angle

AB = 2niL/c@. (9)

Now A8 is produced by the photospheric random walk with velocity I$ and correlation time 7. such that

where rD is the radius in the photosphere. The energy flux density into the corona is

d B; F= A,‘-

J -2~r dr dz,

dt HIT

where A, is the cross-sectional area of the flux tube in the corona; thus Qi = B,,A, where B,, is the coronal field strength. If V, a r,,, i.e. a uniform rotation, we obtain

where V, is V, at the edge of the flux tube and B, is the field in the photosphere. Taking B,, = 50 G, B,=lSOOG, V,=lkms-‘, L=105km,and~=500s,weobtain F=7.5X105ergcm ’ S -I, which is too small by an order of magnitude, although it is adequate for the quiet corona. However, note that this model too contains the desirable feature of an inverse dependence of F on L. Note also that this model does not require specification of some steady-state value of t. as

was necessary in evaluating eqs. (1) and (5). Eq. (12) represents a continual flux of magnetic energy into the corona, which is possibly dissipated as the increasingly twisted flux tube becomes kink unstable [47]. This would be particularly interesting to investigate via numerical simula- tions. This model does not invoke current sheet formation or reconnection, although this could be a consequence of the instabilities; again, simulations would be most interesting. Finally, note the appearance of BP in eq. (12); the necking down of the flux tube in the photosphere plays an important role in this model. and it should not be overlooked in numerical work.

Since the photospheric flux tubes are unresolved, there is little hope at present for observing the postulated twisting. However, the solar spicules tend to show twisting motions, so twists are not out of the question.

A related idea has been proposed by Heyvaerts and Priest 1481, and developed by Dixon et al. [49] and Browning [50]. They postulate that the magnetic energy which is stored in the corona by the shuffling of the photospheric flux tubes is released via reconnection. subject to the constraint that global magnetic helicity is conserved. This scenario has in fact been observed in tokamaks

J. V. Hollweg / Heating of the solar corona 211

[51], but Parker [3.5] has noted that it is not clear how to apply this idea to the corona where the field is presumably twisted and interwoven at random so that the global magnetic helicity is close to zero. The results of an analysis which is beyond the scope of this paper is that the state of minimum magnetic energy is then a constant-a force free field

v xB=aB,

where cx is spatially uniform. Numerical simulations to test this result would be very interesting, and the question raised by Parker should be addressed. This scenario leads to heating rates which scale in much the same way as in Parker’s model. For example, Browning [50] considers a cylinder which is twisted an then relaxes to the minimum energy state on a time scale t,. For a velocity profile

V, = 4?$r( R - r)/R*,

where R is the radius of the cylinder, she obtains an energy flux density which goes into heat given by

F = (4/751r)@Vg( t,/t,) R/L, (13)

where t v is a velocity time scale, t v = R/V& Eq. (13) is qualitatively similar to eq. (l), but smaller by the factor 16/75. The heating rate saturates when t, = t,, so setting these time scales equal optimizes F. Taking B, = 50 G, R/L = 0.05, and V, = 1 km s-‘, we get F = 2 x lo5 erg cm-* S -I which is almost two orders of magnitude too small for an active region. However, Browning did not consider the necking down of the flux tube in the photosphere, so for V, it might be more appropriate to take the larger velocities on the expanded chromospheric part of the flux tube. If V, scales as R near the ends of the tube where the twist is applied (this is the case in the model of Sturrock and Uchida discussed above), then taking V, = 5 km s-’ might be more appropriate (if B,, = 1500 G at the base and 50 G in the expanded part). In that case F = lo6 erg cm-* s-r, which is adequate to heat the quiet corona, but not the active corona.

We thus have a series of models in which magnetic energy is gradually built up in the corona due to the twisting or shuffling of the photospheric flux tubes. This energy is then released via reconnection if current sheets form, or possibly via ideal MHD instabilities. The principle virtue of these models is that they are qualitatively in accord with increasing evidence that the coronal heating is “ bursty”, and that what we see as the corona is really the averaged effect of many small “ flares”. For example, Porter and Moore [52] have observed brightenings in the spectrum of C IV. For a quiet region, they estimate an average input of about 5 X lo5 erg cm-* s-r; this figure is based on there being some lo4 events of 1O26 ergs per event on the sun at a given time, with an average lifetime of about 30 s. This energy appears to be released in low-lying loops, with heights of 2-4 X lo3 km. The energy is adequate to heat the quiet corona, and it should be borne in mind that the observations are for a quiet region. (This energy is still an order-of-magnitude smaller than the required heating of the underlying chromosphere.)

Observations of bursty heating and short-lived dynamic events have to be interpreted with caution, however. After all, waves are inherently dynamic and time dependent, and it is conceivable that the observed temporal variations are associated with waves entering the corona

from below. One might be seeing the effects of impulsively generated waves, or waves which have steepened into shocks in the chromosphere; indeed, Hollweg et al. [53] have shown that even the notoriously linear Alfven wave can form shocks in the chromosphere. If the energy is released in low-lying loops, as in ref. [52], one then has to worry about how the energy gets to the much greater heights of the quiet loops or the larger active region loops. The events observed by Porter and Moore might do this by launching waves to greater heights. This has been suggested by Brueckner [54], who gives a convenient summary of the dynamic jets and turbulent events observed at the base of the corona.

On the other hand, the hard X-ray spikes observed by Lin et al. [55] are almost certainly a signature of reconnection, and not waves, and it is crucial to determine how much energy is actually released in such events.

The models outlined above also have significant failings. We have seen that the derived values of F tend to be significantly less than the 10’ erg cm ’ s ’ required for active region loops. Alternatively, the steady-state value of t required in models such as Parker’s (eq. (1)) must be very long, implying rather large displacements of the photospheric flux tubes; it is not definitely known whether the flux tubes actually get shuffled to the required extent. but video magneto- grams give the impression that they do not [39]. Furthermore, everyone agrees that these models fail on open magnetic field lines. since footpoint displacements simply launch waves which propagate out into the solar wind, and magnetic stresses do not build up. These models also cannot account for the chromospheric heating. In a model such as Parker’s, displacement of an initially vertical field produces a horizontal field component which is essentially the same in the chromosphere and corona. The associated magnetic energy density is thus the same in these regions, and the energy flux density F is proportional to the thickness of the region in question. These models can thus not explain the astonishing fact that the value of F required to heat the very thin chromosphere, with a thickness of some 1500 km, is nearly the same as the value of F

required to heat coronal active region loops with lengths of the order of lo5 km. However, to the extent that acoustic heating of the chromosphere is viable, this may not be a problem after all.

The model calculations described above completely omit the dynamics which must occur when the accumulated magnetic stresses are released into heat, via reconnection or instabilities. Whatever dynamics occurs is bound to be complicated and nonlinear, and the field is ripe for numerical simulations. Attention should be paid to quantitatively assessing the coronal heating rate, the required footpoint displacements, and the magnitude of the coronal motions.

The controversy over whether current sheets and reconnection are inevitable should also be investigated via simulations. An attempt in this direction has been made by Mikic et al. [56]. They considered a box with an initially uniform vertical field. The footpoints at one end were randomly shuffled in a series of steps; the induced dynamical motions were allowed to die out between steps. As time went on, they found that the rms current density increased exponentially with time, and they took this to be qualitative evidence supporting van Ballegooijen’s model, which argued against inevitable current sheet formation. On the other hand, their results do show the gradual development of thin regions of large current density, which mu-~ be the disputed current sheets, but unresolved because of the finite numerical grid spacing. What appears to be needed is some way to decide whether this model supports Parker’s arguments or not. It appears. however, that the large current densities in the simulation grow gradually, which does not seem to agree with Parker’s arguments that current sheets should appear essentially immediately due

J. V. Hollweg / Heating of the solar corona 219

to topological constraints. We note too that Mikic et al. overlooked another prediction of van Ballegooijen’s model, viz., that the rms horizontal field component grows linearly with time. In the numerical simulations, it appears that the horizontal field also grows exponentially with time, but at a slower rate than the current density. We do not know how to relate this to either Parker’s or van Ballegooijen’s models. This simulation obviously contains some interesting new physics which has to be understood quantitatively.

5. Waves

In view of the exciting new ideas proposed by Parker and others, wave models for solar atmospheric heating have come to seem a bit old-fashioned, but waves still have their virtues. They appear to be necessary for heating the regions with open magnetic field lines. Indeed, outward-propagating AlfvCn waves have been observed in the solar wind [57], and they may have been detected at heliocentric distances of 2-10 solar radii by radio measurements of Faraday rotation fluctuations [58], which is the only method at present for detecting magnetic field fluctuations in the corona. The sun does seem to radiate AlfvCn waves with significant energy fluxes, but it is not known whether the waves originate from the convective motions, or whether they are the remnant of reconnection events occurring somewhere above the convection zone. It is also not known why the observed AlfvCn waves have dominant periods of several hours or longer; flux cancellation events or the mesogranulation may determine the time scale. In any event, the presence of waves in the solar wind suggests that wave theories should not be discarded. Waves have other appealing features. If the waves originate at low heights, then they propagate through the photosphere, temperature minimum region, and the chromosphere, and they can in principle heat those regions on their way to the corona and solar wind. Waves also exert ponderomotive forces on the solar atmosphere, and they may be able to produce dynamic phenomena such as the spicules and jets.

Consider a low-beta plasma such as the corona and upper chromosphere. The slow mode can be thought of as being essentially sound waves guided by a nearly rigid magnetic field; the small group velocity means that they cannot supply the coronal energy requirements, and we will not consider them further. The fast mode’s group velocity is adequate to supply the energy, but they tend to be totally internally reflected in the chromosphere, and evanescent in the corona. The reasoning is as follows: The low-beta dispersion relation is approximately

k,2 = u’/v; -k;,

where w is angular frequency, and k, and k, are the horizontal and vertical wavenumbers. In the corona v, is large, and k, is expected to be large if the horizontal spatial scale is associated with the distance between the magnetic flux tubes. The result is that for reasonable values of w, kt < 0 in the corona, implying evanescence. The remaining mode is the AlfvCn mode. It is not totally reflected, and its group velocity is along B, providing a natural association between enhanced heating and the magnetic field. Its WKB energy flux density is

F= p(6V2)vA. (14)

Ifwetakep=5x10P’5gcmP”. uA= 2000 km s -‘. and (6V2) = 2 x (30 km SC’)’ allowing for two degrees of freedom, we find F = 1.8 X 10’ erg crn~-’ s ‘. Thus if the observed coronal nonthermal velocities are Alfven waves (or the closely related Alfvenic surface waves). the energy flux densities are adequate to supply the active region loops. At the base of a coronal hole we take B, = 6 G and p = 3.3 X lop” g cm-‘. giving F= 6 X 10’ erg cm-’ SC’, which is only slightly lower than the requirements for a coronal hole with a high-speed solar wind flow: and in fact it will be shown below that non-WKB effects enhance the energy flux somewhat.

The above discussion is for a uniform medium. Waves on the active region loops arc conveniently thought of in terms of surface waves. To convey the essential physics of a surface wave, and the phenomenon of resonance absorption which will be discussed below. consider an incompressible fluid containing a planar tangential discontinuity which supports the surface wave. We take the wave to move along the background magnetic field, and we work in the frame moving with the wave. In that frame there is a steady flow over the crests and troughs of the deformed wavy surface. The total (fluid plus magnetic) pressure. p,<,‘. must be continuous across the surface. In terms of pt,‘. Bernoulli’s equation is

P’,,~ + p* V2/2 = constant on streamlines. (15)

where

p* E p - B2/4.# = constant on streamlines (16)

and the streamlines are parallel to B. Now the flow accelerates in going over a crest and decelerates at a trough. A crest on one side of the surface corresponds to a trough on the other side. and if the wave amplitude is small the acceleration on one side is equal to the deceleration on the other side. The total pressure balance then requires

py = -p; (17)

where the subscripts 1 and 2 refer to the two sides of the surface. This requires

Since we were working in the frame moving with the wave. V in eq. (18) is the phase speed. Now the fluid is incompressible. and we can add an arbitrary field component perpendicular to k without changing the ptot balance. Thus

V;hhasc = (B,:, + B&&P, + ~21-7477. (19)

where the subscript k indicates the component along k. It turns out that eq. (19) applies also to waves on a cylindrical plasma column if gravity is

neglected, if the background magnetic field is parallel to the axis, if there is no background flow. and if 1 wz 1 /‘R B 1 k, 1, where R is the radius of the plasma column and the B-dependence of the motions goes as exp(im0) with 1 n? I = 1. 2, 3. . .: the plasma may be compressible [59]. If I m 1 = 1, the wave is called a kink mode.

J. V. Hollweg / Heating of the solar corona 221

If the “surface” which supports the wave is not a true discontinuity, but a thin layer of thickness a across which things vary smoothly, then the wave can damp via a process called remnance absorption. In an initial value problem, the global surface wave decays and in ideal MHD the energy which was originally in the surface wave on a spatial scale of k-’ goes into a thin layer of thickness ka2 -=c k -I inside the “surface” [60]. The transfer of energy from large to small scales is reminiscent of turbulence. If some dissipation mechanism is added to the system, then the energy in the thin layer can be converted into heat. This process has been applied to coronal heating, originally by Ionson [61].

Resonance absorption also occurs if a propagating wave impinges on the surface from one side. If the surface is a true discontinuity, then one has a standard reflection/ transmission problem and energy is conserved. But if the “surface” has non-zero thickness, then some energy stays in the surface and the incident energy flux is greater than the reflected and transmitted

fluxes. To get a feel for the physics behind resonance absorption, consider again the incompressible

surface wave, and work in the wave frame. If the “surface” is thin, then the total pressure must still be approximately continuous across the surface; this is equivalent to neglecting the inertia in the surface. Then eq. (17) is still approximately true, implying that there is at least one location in the surface where p* = 0. But Bernoulli’s equation (15) then says that ptot is constant along those streamlines, and it then becomes impossible for the variations in ptot along the surface to be constant across it. Something has gone wrong. The error is our assumption that there is a frame in which a/at = 0. Thus there is no normal mode.

Hollweg [62] and Hollweg and Yang [63] have shown that the resonance absorption can be simply understood if we regard the total pressure fluctuations, 6ptot, as known, and approxi- mately the same as the surface wave on a true discontinuity. If we still consider the incom- pressible case, but now work in the inertial frame (in which there is no background flow), we obtain the following equation for the velocity fluctuation in the direction of k, which defines the x-axis:

d’(k.uA)2) Sv,= -;!!$$_ at2

(20)

This is the equation for a harmonic oscillator driven by the total pressure perturbations associated with the global surface wave. The field lines corresponding to p* = 0 are driven at resonance, and SV, (and 6B,) grow secularly. This secular growth of energy must be accompa- nied by decay of the global surface wave, since total energy is conserved.

Actually, only the p* = 0 field lines are strictly in resonance with the surface wave. But neighboring field lines will be effectively in resonance as long as wt and k,u,,t are approxi- mately in phase up to t = t,, where t, is the surface wave decay time. At later times the surface wave has decayed away, and it does not matter if things get out of phase.

1 w - k,u,, 1 t, < ,rr/2.

Eq. (21) yields the result that the energy ends up in a layer of thickness =

We require

ka2.

(21)

At times long compared to I, the behaviour is given by the homogeneous part of eq. (20). since the driver has decayed away. Each field line in the vicinity of p * = 0 oscillates at its natural frequency. (This is the Al’utn continuum.) Neighboring field lines get increasingly out of phase (this is commonly referred to as phuse mixing) while the energy remains contained within an envelope of constant width. However, in virtue of the phase mixing, strong cross-field gradients build up, leading eventually to dissipation via shear viscosity or electrical resistivity. Thus phase mixing has nothing to do with the buildup of energy near p* = 0 or with the resonance absorption process itself; it has only to do with the heating at later times.

These ideas have been extended to a compressible plasma by Hollweg and Yang [63]. We use their equation (69) to obtain the damping rate in the active corona. For a cold (i.e. low-beta) plasma in a uniform background magnetic field they obtain the following expression for the damping rate due to resonance absorption:

Y/W = ?Tku I PO1 - PO2 I /a PO1 + PO2 1’ (22)

where it has been assumed that k is nearly perpendicular to B, and that the density varies linearly across the “surface”. Note that the damping rate is proportional to the surface thickness. This comes about because more field lines are then effectively in resonance with the driver (eq.

(21)). We take k = l/R, corresponding to a kink mode on a thin cylindrical plasma column [59], po2/po, = l/3. and a = 2000 km, which is 20% of the diameter of a loop having L = 10’ km and D/L = 0.1. If the wave period is 100 s (for the fundamental resonance on a loop with L = lo5 km and w/k,, = 2000 km s-‘) we obtain y-’ = 200 s. This is very close to the required value of 180 s given at the beginning of section 4. Coronal active region loops can be heated by waves which dissipate via resonance absorption [63-661, if the nonthermal motions in the corona are indeed waves.

Note that

y CC aB,k,/R

(using cylindrical coordinates). For active region loops of varying sizes we might expect u/R = constant. Thus the heating will be stronger where the fields are stronger. And if the wave spectrum in the loops is determined by the cavity resonances, we would have k, ti L ‘. and shorter loops will get more heat. These are favorable features of the resonance absorption mechanism.

The question of how the energy is ultimately converted into heat is a thorny one. and we must resort to some speculations. In a system which is continuously driven. the dissipation due to viscosity or resistivity (which we shall collectively denote by VI) occurs in a layer of thickness CC q’,” and the maximum velocity amplitude scale as .q7 ‘I3 [67,68]. For sufficiently small 7, the damping rate is independent of 71, and eq. (22) still holds, as originally speculated by Ionson [61]. The thickness and velocity amplitudes of the heating layer adjust themselves to absorb the energy which is purnped in by the global surface mode, and one can compute the damping rate without specifying the dissipation mechanism. In the corona 17 is small, the heating layer is very thin. and linearity fails. (For example, if we take 77 to be a classical shear viscosity, we find from section 7

J. V. Hollweg / Heating of the solar corona 223

and eq. (116) of ref. [63] that the maximum velocity becomes comparable to the AlfvCn speed if r~ 5 1O-5 in c.g.s. units. This is comparable to Braginskii’s q3 and q4 in the corona. For

comparison, q0 = 1 if Tp = 2.5 X lo6 K, but q0 does not affect the AlfvCn resonance.) Nonlinear

mechanisms may provide the conversion into heat. Strauss [69] has proposed that the region where energy is absorbed may be susceptible to ballooning instabilities, which spread the heat over a larger volume. Hollweg and Yang [63] have suggested that the strong velocity shears in the vicinity of the resonant field line might lead to Kelvin-Helmholz instabilities, which in turn might initiate a turbulent cascade of energy to the small scales required for conversion into heat. Under the assumption that the turbulence proceeds at the Kolmogorov rate, they estimate an eddy viscosity, which is used to estimate the thickness of the heating layer and the maximum velocities; quite reasonable numbers were obtained.

The postulated nonlinear effects pose another problem. The analysis of resonance absorption assumes well-ordered surfaces or regions which satisfy a simple resonance condition. But nonlinear effects such as turbulence presumably tangle up the field and plasma and destroy the assumed order. It is not known whether resonance absorption can proceed under these cir- cumstances, and numerical simulations will probably be required to settle the issue. See, however, Similon and Sudan, ref. [70], for a discussion of issues related to wave propagation in a disordered environment, which is itself an interesting topic for numerical work since analytical results are few and difficult to obtain.

Serious attention should be paid to numerical simulations of the nonlinear effects likely to occur in association with resonance absorption. Do instabilities occur? Do they effectively distribute the heat through a large volume and limit the maximum velocities to values compatible with observations? Do tangled fields develop, and how do they affect the absorption? Some interesting simulations of the linearized incompressible situation have been given by Steinolfson [71], including the effects of electrical resistivity; see [68] for some physical insights into Steinolfson’s numerical results. Poedts et al. [72] have also recently presented some interesting time-dependent linear simulations for a compressible plasma.

Another wave dissipation mechanism is MHD turbulence. Its possible relevance is motivated by the remarkable fact that the Alfven waves in the solar wind have a power law power spectrum, which in lowest order is close to the k- 5’3 Kolmogorov spectrum [73], although there are many fascinating details which are conveniently summarized in the recent paper by Velli et al. [74]. Electron density fluctuations close to the sun also have a nearly Kolmogorov spectrum [75]. How this nearly Kolmogorov spectrum comes about is not known. In any event, the volumetric heating rate associated with a Kolmogorov cascade is

EH = P (6 V2)3/2/LCOrr 3 (23)

where L,,,, is the correlation scale, i.e. the size of the largest eddy. All quantities in eq. (23) can be measured in the solar wind, and it turns out that (23) gives a good approximation to the observed proton heating (see refs. [76,77] for summaries of the data). It turns out that eq. (23) “works” also in the chromosphere [78], spicules, active region loops, and at the bases of coronal holes. For example, in an active region loop we take p = 5 x lo- i5 g cmp3, (6 V’) = 2 x (30 km SC’)‘, and L,,,, = 5000 km (the loop radius); then eq. (23) gives E, = 8 X lop4 erg cmp3 SC’, which is only slightly less than the canonical value of 10e3. At the base of a coronal hole we can

224 J. t’. Hollweg /’ Hearing of the .ro/ar coronal

use the propagation equation for damped AlfvCn waves [79] to deduce a damping Iength

L, = LcorrUA( 6V2) --l’*.

If we take L,,,, = 14000 km, tlA = 900 km s- ‘, and (6Y*) = 2 x (25 km s-‘)~, we obtain half a solar radius for L,; this is the required value deduced by Withbroe [24].

Hollweg and Johnson [79] have produced a model for the corona and solar wind which derives its energy and momentum from AlfvCn waves which dissipate according to eq. (23). Unlike other solar wind models, this one does not assume an already-heated corona as an inner boundary condition. The model succeeds in producing a steep temperature rise to coronal temperatures (although the coronal base pressure is somewhat too low) and moderately high-speed solar wind streams. But it has a major flaw: it predicts much too large proton temperatures near r = 3 solar radii. which are ruled out by observations of resonantly scattered Lyman-alpha radiation. Something has gone wrong. The papers by Velli et al. [74,80] may provide a useful alternative description.

Kolmogorov turbulence and eq. (23) are not the I%- 3/2 dependence, and

En = p{ GV2)2/L,,,,r?j,.

The extra factor of 6Y/oA results in coronal and too small. Remarkably, Kraichnan turbulence has

Turbulence is driven by the nonlinear term

only possibilities. Kraichnan turbulence has a

chromospheric heating not been observed.

rates which are much

(24)

in the momentum equation. For a pure AlfvCn wave in a uniform medium the square bracket in (24) is zero, and there cannot be turbulence. But one does not have this situation in the corona or chromosphere [Sl]. Waves can reflect back and forth on the coronal loops, and they can to some extent be thought of as standing waves, in which case the bracket in (24) is locally non-zero. The same applies in the chromosphere, and non-WKB effects due to the strong AlfvCn speed gradient also make the bracket non-zero. And even short-wavelength waves in non-uniform medium have a non-zero bracket. For example, ref. [82] considered the highly idealized situation of a lineariy polarized AlfvCn wave propagating outward from the sun in a spherically symmetric solar wind with a radial magnetic field. From higher-order terms in a WKB expansion there results

(25)

where V, is the wave amplitude, V, is the solar wind speed, r is heliocentric distance, and the wave varies locally as exp(ikr - iot) with w/k = v0 + uA. The divergence in (24) introduced a multiplicative factor of k, and Velli et al. [74,80] have used dimensional arguments to suggest that (24) and (25) then leads to an inertial range with a k-i spectrum. The heating rate would

J. I/ Hollweg / Heating of the solar corona

then be approximately

225

EH v,+u, du,

= p(W)3’27 dr

It is fascinating to note that the magnetic field power spectrum in the solar wind goes as k-r at low k; on the other hand, almost any function has a k-’ part to its spectrum at low k, so k-’ is not a guarantee that one has a turbulent inertial range. It is not clear how one can make the distinction.

In addition to dissipation, another issue which must be addressed in wave heating models is the wave propagation. It must be demonstrated that sufficient wave energy can actually propagate from the convection zone into the corona, in spite of the strong reflections which occur due to the rapid increase of AlfvCn speed with height in the chromosphere; for example, U, increases from about 7.7 km s-l . m the photospheric flux tubes (B, = 1500 G, p = 3 x lop7 g cmP3) to about 2000 km s-l in the active corona.

Very little believable work has been done in this regard. The strong vertical gradients (due to gravity) and the strong horizontal gradients associated with the flux tubes (which fan out rapidly with height) make the problem analytically untractable, and numerical simulations would be of great use. One can expect not only interesting nonlinear effects to occur as the wave amplitudes increase with height [53,83], but also linear and nonlinear mode couplings as the waves encounter the curved magnetic field lines and regions of varying plasma beta [83,84]. A key question to be asked in numerical computations is how much energy actually reaches the corona, and is it in a form which can be efficiently dissipated? Some analytical studies have been made of the kink mode using the thin flux tube approximation (see the review by Roberts [83]), but the flux tubes are only thin in the photosphere. More work has been done for the AlfvCn mode, which is noncompressive in linearized theory and therefore does not couple with gravity or radiation.

The AlfvCn mode has been modeled as an axisymmetric torsional motion on a cylindrically symmetric flux tube with a vertical axis of symmetry. Since AlfvCn waves carry energy along the background magnetic field, the only spatial parameter in the problem is s, the distance along B,,. This makes the problem one-dimensional and it is possible to do some detailed analysis. It is unlikely that the waves are these idealized twists (although the spicules do show twisting motions), but one does these calculations in the hope that they are generally representative of some of the more complicated wave motions which may actually exist.

In cylindrical coordinates the background magnetic field is taken to have only Y and z components which can be arbitrary functions of position consistent with v l B = 0. The lin- earized equations are:

(27)

(28)

226 J. V. Hollueg / Heattng of the solur mmu

where x = 61//r and r(.r) is the distance from the symmetry axis to the field line. Near the symmetry axis one expects r2L$ = constant and then

(29)

Equation (29) is the one usually studied. If we take ~1~ a exp( s/2h) in the lower solar

atmosphere, and uA.c~~r~~,~~l = constant, the energy transmission coefficient is 4~hu/C~A,,,,,,,,i,, where we have taken 2 h Cc)/cA,,_.orona ec 1. If the magnetic field were not clumped into flux tubes, the transmission would be very small. But the clumping increases the energy which can be transmitted into the corona by increasing h in two ways: the larger AlfvCn speeds in the photosphere increase the transmission by a factor of 2, and the longer field lines near the edges of the expanding flux tubes can increase the transmission by a factor of 225. There is still some controversy. but it appears to us that interesting energy fluxes can be carried from the photosphere into the chromosphere and corona by these waves.

There are other interesting non-WKB effects. One does not have local equipartition between kinetic and magnetic energy, and the WKB expressions 6V a ,oO 1,‘4 and 6B a p’o/” no longer hold. This is particularly important in the chromosphere, where the WKB result predicts a much more rapid increase of velocity with height than is actually observed, whereas full solutions of the AlfvCn wave equation are compatible with the observations [78]. The average Poynting flux is not pO( 6V2) u, and care must be taken in using observations of nonthermal motions to deduce an energy flux, especially in the chromosphere [78,81,86].

As an extreme example, consider waves in a coronal hole and take w = 0 [X7]. Close to the sun the time-averaged (indicated by the angle brackets) Poynting flux turns out to be

F = Po( SV’) ~~/(‘A,,,it. (30)

where cA,cr,t is oA at the Affc,Pnic critical point in the solar wind; in deriving (30) we have used r2B, = constant near the symmetry axis. We now find from energy flux conservation that 6V a B; ‘I2 and 6B a BA/‘, in contrast to the WKB results. In the photosphere and chromo- sphere uA < LJ~,,.~,,. and the Poynting flux is less than the WKB result; this is a consequence of the reflections. At the base of the corona, however, uA > ~~~~~~~~ and the energy flux is greater than the value that would be deduced from measurements of nonthermal coronal motions [ll] using the usual WKB results. The reason is that more energy resides in the magnetic fluctuations than in the velocity fluctuations: we have

for this case. For example. at the base of a coronal hole we take B,, = 6 G and p. = 3.5 X 10 ” g cm-‘, giving l)A = 900 km s _ ’ . In a high-speed solar wind stream we might have rlA,cr,t = 600 km S ‘. and the low-frequency Poynting flux is enhanced by 50%. Typical nonthermal velocities in the corona are 20-30 km ss’ (along the line of sight), and if these motions are waves the energy flux is sufficient to drive high-speed solar wind streams. In reality, neither the WKB nor w = 0 limits apply exclusively, and the actual energy flux probably lies between the two extremes.

J. Y. Ho~lweg / Heating of the solar corona 227

If we take 6Va 3;‘/2, then a 25 km s-i velocity in a 6 G coronal field corresponds to a 1.6 km s-i velocity in a 1500 G photospheric field. The latter value is comparable to solar granular velocities.

Another interesting feature of AlfvCn waves is their ability to excite resonances on closed coronal loops [3,81,86,88]. The coronal part of the field line can act as a resonant cavity because waves can be strongly reflected from the dense chromosphere and photosphere at each end of the coronal loop. If the wave source is in the photosphere at one end of the field line, it turns out that the transmission into the corona is greatly enhanced at those frequencies for which an integral number of half wavelengths fits into the coronal part of the loop, i.e.

where L is the length of the coronal part of the field line and n = 1, 2, . . . At these resonant frequencies a large-amplitude standing wave builds up in the corona. The transmission is enhanced because some of this wave leaks out of the corona, and destructively interferes with the downgoing wave which originated at the source but was reflected on its way to the corona, resulting in almost no reflected wave and a large transmission coefficient. The effect is analogous to antireflection coatings on camera lenses. Thus the coronal loops may be driven at preferred frequencies, rather than at a continuum of frequencies.

Solar spicules may also act as resonant cavities, but in this case the resonance occurs when a odd number of quarter wavelengths fits into the spicule, which is bounded above by the rarefied corona and below by the dense photosphere and low chromosphere.

An interesting feature of the loop resonances is the following: Consider a broadband wave source, and calculate the net coronal heating by doing an appropriate integration under the resonance curve, which has a frequency width determined in part by the coronal wave dissipation rate, y, and in part by the rate of leakage out of the coronal cavity. If y is small, then the net coronal heating is proportional to y. But if y becomes large enough, then the net coronal heating will be independent of y, and one can calculate the coronal heating rate without having to worry about the dissipation mechanism. This behavior results when increasing y leads to a decreased energy density in the cavity, but also to a compensating increase in the width of the resonance curve. This behavior was postulated by Ionson [89], and refined by Hollweg 13,811 and Ionson [4,90,91]. One interesting result is that for each resonant peak the energy flux density which goes into heat then scales as

where P(w) is the power spectrum of the source (ref. [3], eq. (74)). Since w,,, a B,/L, we find a flux density scaling as (B,/L)*. We again have the desirable features of enhanced heating in stronger fields and on shorter loops. Numerical estimates show that the resonances can provide enough energy to heat the active region loops, if they can be excited.

Suppose instead that one has a short coronal loop with resonant frequencies which are too high to be excited. If k is the coronal wavenumber, then the energy transmission coefficient (with no allowance for dissipation) is approximately

T= [Y,(a) - tan(kL)/2nhk]-*, (31)

228 J. V. Holiweg / Heating of the sob corona

where (Y = 2 ha/uA,corona (ref. [81], eqs. (32)-(35)). For example, if L = 4000 km, h = 200 km, and tlA.coronu = 2000 km s- ‘, then T = 0.025 for 27i/w = 100-300 s. If the waves are launched in the photosphere with an rms velocity of 1 km s ~’ in a field of 1500 G, then the photospheric energy flux density is 4.6 x lo9 erg cm-’ s-l if we allow for two degrees of freedom. This figure has to be decreased by the area expansion of the flux tube. Bcorona/Bphotosphrre. If B,,,,,, = 50 G, then the incident energy flux density is 1.5 x 10’ erg cm ’ s-l. Multiplying by T gives 3.9 X lo6 erg cm-2 s-I flowing into the corona. This value would be doubled if there were an equal source at the other end of the loop, and thus very short loops can be powered by waves. The problem is that loops of intermediate length, L = 104-5 X lo4 km, have a small transmission coefficient according to eq. (31), but the transmission-enhancing resonances probably have frequencies which are too high to be excited. Perhaps our estimate of h, which was derived for a vertical field line, is too small; the longer field lines at the edges of the flux tubes imply values for h which are larger by a factor 225. Since T a h* for the intermediate length loops, a larger value of h can make a significant difference.

Thus unlike the “magnetic stress models” in section 4, MHD wave models can deal with heating the chromosphere, the photospheric magnetic flux tubes themselves, and the open field regions. If the loop resonances are excited they can account for the heating of the longer active region loops. If the resonant frequencies are too high to be excited, waves can account for heating the very short loops, but there is a problem with getting enough energy into intermediate length loops (unless we have underestimated the quantity h). There are motions in the chromosphere and corona which may be waves. One problem here is that in a low-beta plasma the fast, AlfvCn, and Alfvenic surface waves have velocity fluctuations nearly perpendicular to B,, while the observations show a nearly isotropic distribution of velocities in the corona [92]; in the chromosphere the horizontal velocities exceed the vertical velocities by a factor of about 2 [93], which may be indicative of the effect of the magnetic field. Perhaps the field-aligned velocity fluctuations are simply slow waves which are energetically uninteresting in the corona, or they might be associated with the spicules. Perhaps the isotropy is a result of nonlinear wave dissipation by turbulence or ballooning instabilities. Or perhaps the nonthermal motions are the consequence of the reconnections and nanoflares discussed in section 4, and have nothing to do with waves at all; but then one has trouble explaining why there are motions on open field lines which should not be subject to the buildup of magnetic stress as envisioned by Parker and others. Finally, the chromospheric and coronal motions may be the result of flux cancellation events in the photosphere; no work has been done to explore this possibility.

The simple wave propagation models based on eq. (29) might be grossly misleading if nonlinear effects are important, and here simulations have much to contribute. Hollweg et al. [53] have used a nonlinear FCT code to show that even simple torsional AlfvCn waves can form shocks in the chromosphere. This computation was purely one-dimensional, and an extension to two dimensions, including cross-field gradients would be useful. On the other hand, if solitons can form [95-971, then shock formation would be hindered and the shock-associated dissipation would not occur; this too is a subject ripe for numerical study. An et al. [98,99] have recently investigated eq. (29) numerically. They demonstrate the tendency of the waves to be reflected by the AlfvCn speed gradient, and they also point out that the partial trapping of the waves due to the reflections can enhance the ponderomotive force exerted by the waves on the plasma. In the future, attention should be given to eq. (28), both analytically and numerically.

J. V. Hollweg / Heating of the solar corona 229

The flux cancellation events observed in photospheric magnetograms may also have important consequences for the dynamics and heating of the solar atmosphere. No work has been done in this regard. Since these events are likely to be highly nonlinear, numerical simulations would appear to be essential.

6. Conclusions

We have seen that there are two types of theories for heating the solar atmosphere: wave theories, and theories which involve the gradual buildup of magnetic stress with probably impulsive releases of that energy via reconnection. Both types of theories are observationally motivated.

In the solar wind one sees a ubiquitous field of AlfvCn waves which are probably of solar origin. They appear to dissipate via turbulence, and heat the protons and heavy ions. (The heavy ions, such as He2+, are particularly intriguing. They tend to have Ti/T, = mi/m,, where T is temperature and m is ion mass. They also flow faster than the protons, with v - VP = u,. The latter observation almost certainly implicates the waves. The fact that the heavy ions can have temperatures in excess of the coronal temperature clearly demonstrates that heating is occurring in the solar wind, and the mass dependence can probably be explained only in terms of waves.) The solar wind seems to be one part of the corona which is being heated by waves. There is no evidence for bursty heating or reconnection, but there are tangential discontinuities (of unex- plained origin) with an occurrence frequency of 0.1-1.0 h-‘, and rotational discontinuities which occur at a rate of about 1 h-‘; the rotational discontinuities probably evolve nonlinearly from the AlfvCn waves.

There is also some evidence supporting the existence of the coronal loop resonances [loo-1021, but these scanty data hardly constitute a smoking gun.

The principal difficulty with the wave theories is getting enough energy into the corona. On the other hand, observations of coronal loops and bright points increasingly point toward

bursty heating, possibly associated with the magnetic stresses, current sheets, and reconnections postulated by Parker. These data should keep these models alive, in spite of their difficulties in the chromosphere and open field regions; keep in mind however that acoustic heating of a substantial part of the chromosphere is still a possibility.

Thus the kind of model one likes depends in part on which data one looks at. The solar wind data give a picture of a rather smooth plasma with a continuous field of AlfvCn waves having most of their power at long periods (hours) and large spatial scales (0.01-0.1 AU). The tangential discontinuities seem rather benign. On the other hand, X-ray and ultraviolet data imply the importance of structures, localized energetic events, and impulsive events. In some ways the corona resembles “silly putty”, which under some circumstances can be deformed smoothly, while at other times it cracks and forms discontinuities. It seems probable that both types of theories, summarized in sections 4 and 5, have a proper place. There is, after all, no rule of nature requiring that the solar atmosphere be heated by a single process. Both types of theories have their successes and failures, and where one fails the other may save the day. And perhaps the flux cancellation events drive the atmospheric heating, directly by reconnection and/or by launching waves; this possibility needs serious study in the future.

Acknowledgement

Over the years the author has benefitted from discussions with virtually everyone working on solar atmospheric heating, but special thanks go to S. Antiochos, M. Berger, P. Browning, M. Goossens, A. Hood, J. Ionson, M. Lee, S. Martin, E. Parker, S. Poedts, J. Porter, E. Priest, B. Roberts, R. Rosner, A. van Ballegooijen, M. Velli, and G. Withbroe. This work was supported in part by the NASA Solar-Terrestrial Theory Program under Grant NAGW-76, and in part by NASA Grant NSG-7411.

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