JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 91, NO. A4, PAGES 4111-4125, APRIL 1, 1986

Transition Region, Corona, and Solar Wind in Coronal Holes

JOSEPH V. HOLLWEG

SpaceScience Center and Department of Physics, University of New Hampshire,Durham

Previous wave-driven solar wind models (Hollweg, 1978) have been extended by including a new hypothesisfor the nonlinear wave dissipation.The hypothesisis that the waves dissipatevia a turbulent cascade at the rate given by (1) and the waves evolve according to (16). A subhypothesisis that the relevant correlation length scales as the distance between magnetic field lines. This hypothesis allows us to treat the corona and the solar wind on an equal footing; unlike in previous wave-driven models, we do not assumethat the coronal heating takes place below the base of the model. The models exhibit the correct qualitative features, viz., a steep temperature rise (the transition region) to a maximum coronal temperaturein excessof 106 K, and a substantialsolar wind massflux in excessof 3.5 x 108cm-2 s-x at 1 AU. However, the model fails in detail. Parameters that yield a high-speed flow at 1 AU have base pressuresthat are too low; parameters that yield correct base pressureshave low solar wind flow speeds. However, the model "comes close." Thus although we have not shown that the initial hypothesis is consistent with available data, we feel that there are sufficient uncertainties both in the model and in the data to precludeoutright rejection of the hypothesisaltogether.

1. INTRODUCTION Wragg and Priest [1981] have considered the effects of en- thalpy flow, but the heat input was specified ad hoc, without Historically, the origin of the solar wind has generally been physical motivation. Earlier models by Gabriel [1976] includ- treated separately from the origin of the hot corona. Be- ed enthalpy flow, but all heat input was implicitly assumedto ginning with Parker's [1958] original treatment, virtually all occur above the upper boundary of the model. Vanbeveren solar wind models to date start at a lower boundary, where and de Loore [1976] also included the enthalpy flow in a study the corona is already hot. No explicit coronal heating terms of coronal heating by acoustic waves and shocks, but their are included, and it is thus implicitly assumed that all coronal study did not emphasize the conditions found in coronal heating has taken place somewherebelow the lower boundary. The energy of the hot corona is made available to drive the holes. In thesestudies, the solar wind massflux is simply given solar wind. In some models [e.g., Durney 1972, 1973; Durney as a boundary condition, and no attempt is made to self- and Hundhausen, 1974; Durney and Pneurnan, 1975; Hartle and consistently determine the mass flux via the requirement that Sturrock, 1968; Holzer, 1977; Holzer and Leer, 1980; Kopp the flow pass through the Parker critical point; indeed, Wragg and Holzer, 1976; Nerney and Barnes, 1977, 1978; Parker, and Priest neglectthe effectsof flows in the momentum equa- 1960, 1963, 1964; Steinolfson and Tandberg-Hanssen, 1977; tion altogether and instead use the equation of hydrostatic Whang and Chang, 1965] the solar wind energy is derived equilibrium. (Note that the above-cited studies, which include almost entirely from the conduction of heat out of the hot the outflow of enthalpy in a coronal hole, should not be con- corona. In other models [e.g., Alazraki and Couturier, 1971; fused with studiesof the effect of downflows through the tran- Barnes et al., 1971; Belcher, 1971; Fla et al., 1984; Habbal and sition region, such as Pneuman and Kopp [1977, 1978] and Leer, 1982; Hollweg, 1973, 1978, 1981a; Jacques, 1978], MHD Wallenhorst [ 1982].) waves supplement and sometimes dominate the heat conduc- In contrast, a number of authors have realized that if the tion energy, but they dissipateso slowly that they make only a enthalpy flow is important in the energy balance of the transi- negligiblecontribution to the heating of the corona per se. tion region and low corona, then a self-consistenttreatment Similarly, in many studiesof the heating of the low corona must satisfy the requirement that the flow pass through the and chromosphere-coronatransition region, the solar wind is critical point. This requirementhas been satisfiedin the model ignored altogether. To some extent this is not surprisingbe- calculations of Rosner and Vaiana [1977]. That paper was causethe coronal and transition region heating problem ante- primarily interested in the extent to which a simple coronal dates Parker's discoveryof the solar wind. See Kuperus [1969] model was able to reproduce observed radio and EUV line for a review of early studies of coronal heating; see also emissionsfrom the base of a coronal hole; the paper contains Flower and Pineau des Forets [1976] and Flower [1977] for an excellent summary of the data. Accordingly, the heat input recent studies of coronal heating by acoustic shocks, and Mc- in the corona was specifiedad hoc. Also, the properties of the Whirter et al. [1975, 1977] for a recent study of heating by solar wind that emergesfrom the coronal hole were not stud- viscous and heat conduction damping of sound waves. ied in depth. A similarly motivated investigation has been Coronal holes, however, are known to be sources of high- reported by Pineau des Forets [1979], except that he examined speed solar wind streams; see, for example, the reviews by the specific case where the heat input is due to a periodic train Bohlin [1977], Hundhausen [1977], and Zirker [1977]. The of strong acoustic shocks; again, little attention was paid to mass flux in coronal holes is significant,and so is the enthalpy the solar wind part of the solution. flux in the low corona and transition region. This latter fact The importance of treating the transition region, corona, has led to a number of studies of the effects of enthalpy flow and solar wind on an equal footing has been emphasizedby on the energy balance of the corona and transition region. Couturier et al. [1979, 1980] and Souffrin [1982]. Couturier et al. [1980] write, "The heating of the solar corona and the solar wind phenomenon are basically related; however, the two parts are generallymodelled independently: the modelsof Copyright 1986 by the AmericanGeophysical Union. the transition zone and corona are restricted to levels lower Paper number 5A8759. than the temperature maximum and the solar wind models 0148-0227/86/005A-8759505.00 begin above it." These authors present a model based on

4111 4112 HOLLWEG.'TRANSITION REGION, CORONA, AND SOLARWIND acoustic shock heating, which self-consistently extends from the observed coronal motions cannot carry the required the transition region out to 1 AU. A special feature of this energy fluxes if they are sound waves or acoustic shock trains. work is its cognizance of the importance of an outer boundary Recentwork has emphasized •the r01 e of themagnetic field in condition on the thermal energy equation; the boundary con- coronal heating. See some recent reviews by Kuperus et al. dition T(r • oo)• 0 has been imposed. A cruder study of the [1981], Hollweg [1981b, 1983], Parker [1983], and Wentzel relationship between coronal heating and solar wind flow was [1981]. See, however, Ulrnschneiderand Bohn [1981] for a presented some years ago by Leer [1980], but the coronal differing opinion. heating was specifiedad hoc, and no attention was paid to the One of our goals in this work.is to provide a new model of behavior of temperatureat large distancesfrom the sun. the transition region, corona, and solar wind in a coronal Surprisingly, some of the most extensivestudies of the re- hole.This modelwill be fully self-consistentin its satisfaction lationships between the transition region, corona, and solar of boundaryconditions at the criticalpoint and at infinity. wind have been pursued by the stellar physics community. But our study will go beyond previousmodels by attempting Hammer ['1982a, b, c, 1983] has studied the heating of stellar to includewhat may be important modifications to the"tradi- coronae with proper consideration of the requirements that tional physics." T(r • oo)• 0 and that the flow passthrough the Parker criti- In the solar wind beyond 10 Rs we will use the "col- cal point. For the most part, the heat input has been specified lisionless"description of the electron heat conduction flux, ad hoc, but his work systmaticallyshows how the coronal and first postulated by Hollweg [1974, 1976]. This form of the heat solar wind properties depend on the magnitude and spatial conduction automatically yields T(r--• c•)--,O, and it also location of the heating. In particular, Hammer emphasizesthe automatically includes effects of the Parker spiral. Closer to qualitative as well as quantitative importance of the "charac- the sun, inside 10 R s, we use the classicalformula for electron teristic damping length" of someputative energyflux coming heat conduction. The two regions are joined discontinuously, into the corona and transition region from below. Indeed, the in the expectationthat the resultsso obtainedare not drasti- resultsof this presentpaper, although quite differentin detail cally different from the actual behavior of a plasma that from Hammer's, confirm the importance of the "damping smoothly changesfrom collision-dominatednear the sun to length." nearlycollisionless far from the sun. This procedure has al- Modelsin a similarspirit, but for the specific case of heating readybeen used by us [Hollweg,1978] and by Holzerand by a weak train of acoustic shocks, have been presented by Leer ['1980], and it has bean shown to yield no contradictions Hearn and Vardavas ['1981a, b] and Vardavas and Hearn with observations.In particular, it yields coronal temperature [1981]. These papers deal primarily with stars other than the profiles that are "reasonable." However, it should be noted sun, but they do pay proper attention to the requirements that that it is not unanimouslyagreed that this is the correct pro- the flow pass through the Parker critical point and that cedure. In fact, becausethe fi-ee paths of the heat-carrying T (r --• oo) --• O. electronsare not small in the corona, there is really no reason All of the above-cited models suffer a number of physical to believe that the classical formula should be valid. See Olbert difficulties vis-a-vis the real solar wind. [1983] and Scudderand Olbert [1983] for an alternate view 1. They assume uninhibited classical heat conduction but one that is less amenableto global model calculations throughoutall space.The effectof the spiral interplanetary suchas we are presentinghere. However,it shouldbe noted magnetic field is not included. Perhaps worse, no attempts that Shoub[1982] has investigated the downwai'delectron have been made to consider departures from classical heat heat conductionin the ti'ansition region and found deviations conduction, due to the weak collisionality of the rarefied solar from the classicalvalues by no more than 40%. wind. For the coronal and solar wind heat input, we will resort to 2. Virtually all the heat input occursvery closeto the sun. Alfven waves of solar origin. We will assume that the waves No heating of the distant solar wind, by MHD waves for propagate in the WKB (small wavelength) limit, and within example, has been included. Indeed, Hammer's models ex- that approximation the momentum addition of the waves can plicitly excludeheating beyond the critical point. Similarly,no be handled fully self-consistently.(The WKB approximation model allows for momentum addition, by MHD waves, for breaks down close to the sun. The wave momentum addition, example,in the distant solar wind. The dynamicimportance of in fact, turns out to be small there anyway, but the breakdown both heating and momentumaddition has been explicatedin of WKB may have important implications for coronal and general terms by Holzer [1977] and Leer and Holzer [1980]. transition region heating' see section 5.) Specificphysical mechanismsand correspondingmodels have The collisionlessheat conduction and Alfven wave pressure been reviewed by Barnes [1979, 1983] and Hollweg [1978, have already been included in detailed solar wind models 1981a]. The omission of momentum addition in the distant [Hollweg, 1978]. In that work the wave dissipation and solar wind may accountfor the low flow speed,(240 km s-•) plasma heating was postulated to occur only when the waves obtained at 1 AU in the model of Couturier et al. [1980]. became nonlinear. This occurs far from the sun, beyond 3. In the region of heating close to the sun, momentum roughly 0.1 AU. Thus the earlier models included heat input addition has been handled inconsistentlyat best. Such mo- into the distant solar wind, but no coronal heating whatso- mentum addition is always associated with waves or shock ever.Accordingly, it was necessaryto assumethat the coronal trains [e.g., Jacques, 1977]. But many of the above-cited heating had already occurredbelow the baseof the model. models omit it altogether, while in other models it is treated In the presentpaper we wish to explore the consequencesof only approximately, even when the authors have a specific a new hypothesisconcerning the dissipation of Alfven waves. heating mechanism,such as weak acousticshocks, in mind. The hypothesis,which will be discussedfurther below, is that 4. Finally, the available models either make no specifi- the plasma volumetric heating rate is cation as to the physicalnature of the coronal heating mecha- EH = FP(C•V2)3/2/L.... (l) nism, or they invoke acousticshocks that are probably not responsiblefor the heating. We do not wish to provide a wherep is massdensity, (6v '•) is the velocityvariance associ- detailed review here, but the essenceof the argument is that ated with the wave field, and L .... is a measureof the trans- HOLLWEG.' TRANSITION REGION, CORONA, AND SOLARWIND 4113

verse correlation length. The quantity F is a factor of the However, it should be noted that it was recognizedat a very order of 1, which we shall henceforth ignore by absorbing it early stage that the wave field could not be described as a into L .... . (It should be made clear that the waves themselves linear superpositionof small-amplitudenoninteracting waves. are here being regarded as the source of the heating. Thus This is implied by the large amplitudes of the waves and by (6v:) shouldonly includethe powerassociated with the Alf- their nearly noncompressivenature. Early nonlinear attempts venic fluctuations' power associated with other structures, to deal with this aspect were made by Barnes and Hollweg such as high-speedsolar wind streams,should not be included. [1974]. To this day, the noncompressivenessof the waves re- In other words, we are seekingan energy sourcefor the high- mains unexplained,although the efficientLandau-transit time speedstreams, rather than regardingthe streamsthemselves as damping of compressivewaves (see the summary by Barnes a sourceof energy. Similarly, L .... should be regardedas the [1979]) undoubtedlyplays a role. But the noncompressiveness correlation length of the Alfvenic fluctuations.)Although our does suggestthe importance of nonlinearity. use of (1) is motivated by in situ observationsat 1 AU, it is The observed minimum variance direction of the magnetic perhapsremarkable that it seemsto provide a reasonablerep- field fluctuations in the solar wind coincides quite well with resentationof the heatingof the chromosphereand solar coro- the direction of the Parker spiral. This fact too is at variance nal active region loops as well (seebelow). One of our goals with the predictedbehavior of a linear superpositionof nonin- here will be to use a simple model to ask whether (1) can teracting waves,as has been emphasizedby Barnes [1981] and explain the heating of the corona in coronal holesas well. Our others. answerto this question will be a qualified "yes"'(1) can indeed But most important has been the long-recognizedfact that yielda transitionregion, coronal temperatures in excessof 106 solar wind power spectraare typically power laws over a wide K, and hot high-speed solar wind flows. But we will find that range of wave numbers. As was emphasized by Coleman there are still some detailed discrepancieswith observations. [1968], this is reminiscent of the inertial range of hy- This may mean that (1) is inadequate as a complete repre- drodynamic turbulence. This fact, coupled with the other sentation of the entire region between the transition region above-mentioned indications of the importance of nonlin- and 1 AU. But it is also possiblethat some of the discrep- earity, has led to a recent interest in using the language of ancies result from the procedural approximations of the turbulence to describe the fluctuations in the solar wind. See model. recent papers by Barnes [-1981, 1983], Dobrowolny et al. The plan of the paper is as follows.In section2 we motivate [1980], Matthaeus and Goldstein [1982a, b], Montgomery our use of (1). In section 3 we present the equations of the [1983], and Tu et al. [1984]. It has even been suggested model and the method of solution.In section4 we presentthe [Matthaeus et al., 1983] that "the frequent observation of numerical models along with some general comments con- outward-propagating Alfvenic fluctuations in the solar wind cerning how they relate to observations.Finally, in section 5 can arise from early states of in situ turbulent evolution, and we summarizethe resultsand speculatehow the model might need not reflect coronal processesas has hitherto been as- be modified to yield better agreementwith observations. sumed." In this paper we shall assumethat the wave and turbulence 2. EQUATION(1)' MOTIVATION conceptsare not incompatible.Indeed Barnes [1981] has em- Since the pioneering study by Belcher and Davis [1971], it phasized,"The wave and turbulencepictures are not mutually has been widely believedthat the solar wind is almost always exclusive.... Any real difference between the two viewpoints populated by Alfven waves, with their senseof propagation lies in the nonlinear development of the fluctuations." We such that they carry energy along the interplanetary magnetic shall assumethat it is still meaningful to talk about a wave field away from the sun. This seemsto be the casewherever in field that propagatesenergy (and momentum) in the manner situ measurements are available, and there is no reason to of Alfven waves. But we shall hypothesizethat the wave field believe that similar measurements taken elsewhere would not dissipatesby turbulently cascadingits energy to high wave lead to the same conclusion. Belcher and Davis based their numbers, where it is absorbedby the plasma as heat. We shall conclusion •011ClatlOll oetw•n v½ further hypothesizethat the turbulent cascadeproceeds at the magnetic field fluctuations, which were found to be consistent rate given by (1). This rate is the same as the Kolmogorov with Alfven waves. The nearly noncompressivenature of the cascaderate for homogeneous,isotropic turbulence in liquids fluctuations was taken to be corroborating evidence.Other [e.g., Batchelor, 1953; Kolmogorov, 1941a, b; Obukhoff, 1941]. facts are consistent with the wave interpretation. If However, it should be clear that the arguments that lead to (1) E + v x B/c = 0 (E and B are electric and magnetic fields, v is in ordinary fluids cannot be immediately applied to the solar plasma velocity, and c is speed of light), then the observed wind, which is compressibleand anisotropic. In the present velocity and magneticfield fluctuationsimply a time-averaged context,(1) is to be regardedas no more than a hypothesis,to Poynting flux in the plasma frame, as would be expectedfor which, however, we can apply some empirical tests. The es- waveswith a preferredsense of propagation.And observations senceof our hypothesisis summarizedby (1) for the cascade [e.g., Bame, 1983' Neugebauer, 1981] indicating that heavy rate and by (16) for the dissipatingwave field (seebelow). ions in the solar wind tend to flow faster than the protons by The motivation for this hypothesisis empirical. As we have nearly the Alfven speed strongly suggest that the ions are seen, the solar wind seems to contain a wave field with a "surfing" on propagating waves (see the theoretical review by turbulent spectrum.We do not offer a physicaltheory in sup- Isenberg [1983]). Observations of Faraday rotation in the port of this hypothesis.However, we note the following points. corona have been shown [Hollweg et al., 1982] to be consis- 1. If the wave field were perfectlyAlfvenic in the senseof tent with predictions based on the idea that the fluctuations 6v = _+6B(4np)-•/:, thenthe nonlinearterms that leadto tur- seenby spacecraftare indeed Alfven waves.Finally, rotational bulencewould vanish,and there would be no turbulence [Do- discontinuities are known to be abundantly present in the browolnyet al., 1980]. Thus the wave field cannot be perfectly solar wind [e.g., Neugebaueret al., 1984], and the most likely Alfvenic. This in fact is observed to be the case in the solar explanation for their origin is that they nonlinearly develop wind, although the reasonsare not understood.Thus designat- from Alfven waves. 4114 HOLLWEG'TRANSITION REGION, CORONA, AND SOLARWIND

3 x104 104 T (K) the mean field. Subsequently,Hollweg [1984] suggestedthat ooo ooo the Kelvin-Helmholz instabilities could initiate and mediate a turbulent cascadeto ever-higher transversewave numbers, kñ. We envision large eddies rubbing together to make smaller eddies via a Kelvin-Helmholz instability, and those eddies in 10-• _ turn rubbing against each other to make still smaller eddies, and so on. Hollweg further noted that if the growth rate [tJv[/Lzcan be identified with the cascaderate, then (1) will result;the correlationlength Lcorrshould then be construedas the transverse correlation length. Finally, Tu et al. [1984] have begun the development of a formalism for treating the turbulent dissipation of a field of waves that in lowest order are Alfven waves propagating outward from the sun. They derive our (16) for the evolution of the waves but with an expressionfor E n that differs from (1). Their expressionfor E n _ is obtained via dimensional analysis,after comparing the sizes - of selected terms in their dynamical equations. However, the terms selected by Tu et al. are not the only possibility. As P (104K ) = 0.33 dynescm -z pointed out by C. Tu (private communication, 1985), some of _ theseterms (for example,the first on the left-hand sideand the - - , , I , , , • I , , , , I , , , first on the right-hand side of equation 16 of Tu et al.) yield a •o• •5oo •ooo 500 time scaleL•/l•vl. If this time scaleis used in the dimensional h (kin) arguments of Tu et al., then (1) results. Thus our basic equa- , ...... I ..... ,,,I ...... I ...... I tions, (1) and (16), are compatible with the formalism of Tu et II 12I 5 • 3 5 10-4-2 10-3 j0-2 j0-1 re(gem ) al., and we refer the reader to that paper for the details of the formal development. Fig. 1. We have reproduced Figure 14 (model F) of Hvrett [1981]. These arguments are weak. So we plan now to further justi- The added solid circles represent the predicted volumetric heating fy our use of (1), by comparing its predictions with rate if (1), viz., Kolmogroff turbulence, is operative. In calculating the solid circles we have used VAL densities, Canfield and Beckors' hori- chromospheric,coronal, and solar wind data. zontal "turbulent" velocities,and L .... = 800 kin. Equation (1) has been examined in the chromosphericcon- text by Hollweg [1985]; we refer the reader to that paper for details.In the chromosphere,p and (tJv2) are known,as func- tions of height, from optical observations. (We have used ing the waves in the solar wind "Alfven waves"is only a values of p from the VAL solar model atmosphere [Vernazza lowestorder approximation.However, note that in the corona et al., 1973]; the valuesof (Sv2) are from Canfieldand Beckers and lower solar atmosphere,non-WKB effectsand the possi- [1975].)The quantity L .... mustbe guessed. Our guessis that bility of having wavespropagating in oppositedirections (as L .... is closely related to the mean distance between photo- on activeregion loops, for example)will in generalyield 5v spheric magnetic flux tubes. The idea here is that the flux 6B(4rrp)-•/2,and turbulenceis possible. tubes act as independentsources of waves; sincethey are inde- 2. According to Table 1 of Montgomery [1983], energy pendent, their separation will be a measure of the correlation can cascade to high wave numbers in two- and three- scaleof the wave field. If we take Lcorr= 800 km and use the dimensional MHD turbulence. observedvalues of p and (tjv2), we can use(1) to predictEn 3. Even if energycascades to high wave numbers,the rate as a function of height in the chromosphere.The results are of the cascadeis not uniquely known in MHD, i.e., (1) is not the solid circles in Figure 1, which have been overlaid on the the only possibility[Grappin et al., 1983]. But there are three, observedheating rate in the chromosphere(taken from Figure rather nonrigorous,reasons for invoking (1). The first is the 14 of Avrett [1981]). The predictions based on (1) agree re- observedform of the power spectrum.Matthaeus and Gold- markably well with the observations.(The bump in the ob- stein [1982a] have examinedmagnetic field power spectrafor servedheating rate in the left side of Figure 1 is probably due three regionsbetween 1 and 5 AU. They note a strongtend- to heat conduction from the hot corona. The decrease in heat- ency for the "inertial range"parts of the spectrato behaveas ing at the right end of the curve might well be an artifact of k -5/3. If the cascadeproceeds at the rate given by (1), and if the inability of Avrett's model to handle the stronghorizontal standard argumentsbased on statistical similarity are em- ployed,then a k-5/3 spectrumcan be deduced.Thus the ob- structuring,due to the flux tubes,at low heights.) servedk-5/3 spectrumis consistentwith (1). But it is not clear Equation (1) can be examined in the active region corona that theargument can be turnedaround, i.e., a k-5/3 spectrum by making use of a data set provided by Golub et al. [1980]. does not uniquelyimply (1). Moreover, the spectrumis prob- This data set provides pressurep, length L, and magnetic field ably not universallyk-5/3. The spectrareported by Bavassano strength B for 15 coronal loop structures.By combining these et al. [1982a,b] are generallynot k -5/3, but they do look data with the loop theory of Rosner et al. [1978], it is possible to deduce the average volumetric heating rate and density of turbulent in the senseof exhibiting what appears to be an each loop, viz., extended inertial range that is nearly a power law in k. The second reason is heuristic. Hollweg [1981b, c] suggestedthat En = 1.8 x 105p•'•?L-ø's3 (2) the shearing motions associatedwith Allyen waves could be unstable to Kelvin-Helmholz instabilities, as long as p = 5.5 x 10-'2(p2/L)'/• (3) _+tJB(4•rp) -•/2. Heyvaertsand Priest [1983] developedthis idea further and showedthat the growth rate is of the order of (all unitsare cgs).For L½or•,we againguess that it is relatedto I&l/Lz, where La_is a characteristicscale length transverseto the mean flux tube spacing,which is in turn related to the HOLLWEG' TRANSITIONREGION, CORONA, AND SOLARWIND 4115

averagefield strength[Spruit, 1981]. We have taken 3. EQUATIONS AND TECHNIQUES L.... = 7520B-•/2km (4) With the exception of (1), the physical assumptionsof this model are very similar to those used by Hollweg [1978, if B is measuredin Gauss.Now if we regardEH and p as 198la]. Briefly recapitulating, observedquantities, and if we believe(4), we can ask how 1. The steadyflow is radial with outward speedv. large the coronal turbulent velocities would have to be if the 2. The cross-sectional area of a stream tube is identical to heatingactually takes place in accordancewith (1). Combining that given by Munro and Jackson [1977] for a specific polar (1)-{4) yields coronal hole, except that we take the net nonradial expansion factor fmax= 4. Munro and Jacksonfound fmax= 7.26 for the {•/•rms= 2.1 x 108(p/LB)1/6 (5) entire cross-sectional area of the hole, but we feel that the for the requiredrms velocity(all units are cgs,and we have smaller value may be more representativeof the central radial taken(6v 2) = 26Vrms2 since there are two degrees of freedom).stream tubes. Table 1 gives{•/•rms (and other salient data) for the 15 loopsin 3. The magnetic field is taken to be radial inside 10 Rs, the data set.We seethat 6Vrmsis in the range 18-30 km s-1 and to follow the Parker spiral beyond 10 R s. All calculations This is generallyconsistent with observationsof coronal non- are done in the equatorial plane. Thus for r > 10 R s, thermal velocities, which are of the order of 10-30 km s-1 [e.g.,Cheng et al., 1979].We thereforeconclude that the hy- BrA = const (9) pothesisthat (1) is valid, when examinedin the context of the and data of Golubet al. [1980], leadsto no conflictswith what is known about nonthermal motions in the corona. See also B, 2 = (rflBr/v)2 (10) Hollweg and Sterling[1984]. whereB• and B, denotethe radial and azimuthalcomponents, Equation (1) can also be examined in the solar wind itself. r is heliocentricdistance, A(r) is the stream tube area, and fl is We definea waveenergy damping length the solar angular rotation frequency.At the solar surface we take Ldamp= Fw/EH (6) whereFw is theenergy flux density. Taking Fw = p(6v2)Vsw, B•s = 1.5fmaxG and inserting(1), yields which, combined with (9) and (10), yields a field strength at 1 Ldamp= L .... Vsw(6V2)-•/2 (7) A U that agreeswith observations. 4. Inside 10 Rs we use the classicalexpression for the Now Mattlaaa•teand ('.n/detainF1QR9a'I ropnrt Lcor r _• f• !AII •'•;• heat conduction flux' at 1 AU. If we alsotake Vsw• 400 km s-• and {6v2)•/2 • 40 kms-• at 1AU, then (7) predicts Ldamp • 1 AU. This estimate q = -KoTe 5/2dTe/dr (11) agreeswith a measuredvalue for thedamping length given by with K 0 = 8.4 x 10- ? (in cgsunits). The electrontemperature Bavassanoet al. [1982b, Figure 5]. Again, (1) seemsto be in is denoted Te. Beyond 10 Rs we assumethat q can be repre- agreement with the data. sentedby what we have called the "collisionless"heat conduc- However, the data of Bavassanoet al. [1982b] have also tion: beenstudied in the interestingpaper by Tu et al. [1984]. The philosophyof their work is very similarto the philosophyof q = •peV• (12) the presentstudy, but their heatingrate differsfrom our (1). Ignoringfactors of order 1, the modelof Tu et al. implies where Pe is the electron pressureand 0•is a factor of the order of 1. In this paper we take o•- 1, but we have verified that the results are not very sensitiveto •. (Note that this procedure differs slightly from Hollweg [1978]. In that paper an attempt was made to self-consistentlydetermine the location of the wherev A is the Alfvenspeed, B(4z•p)-•/2. Equation (8) differs from (1) by a factor{(•¾2)l/2/t2A. Since this factoris closeto unity near 1 AU, Tu et al. also obtain agreementwith the observationsof Bavassanoet al. [1982b]. We concludethat TABLE 1. Observed X Ray and Magnetic Field Parameters observations at 1 AU are not a reliable indicator of the validi- Date ty of either(1) or (8). Note, though,that ((•V2)I/2/t•Ais very Region (1973) L B p En •J/)rrns small in the chromosphereand corona.Thus (8) would not yield the favorablechromospheric and coronal resultsfound AR 375 June 9 1 x 10 lø 34 0.87 6.2x 10 -'• 24x 105 in Figure 1 and Table 1. AR 417 July 4 2 x 10 TM 196 4.0 2.1x 10 -3 20x 105 AR 460 July 31 2 x 10xø 46 2.2 1.3 x 10 -3 25 x 105 In conclusion,even though the physicalarguments support- AR460' Aug. 1 2x 10xø 75 1.5 8.2 x 10 -'• 21 x 105 ing (1) are weak,there are empiricalreasons for believingthat AR 461 Aug. 1 5.0 x 10xø 34 1.2 2.9 x 10 -'• 20x 105 (1) "works" in the chromosphere,active corona, and solar AR461' Aug. 2 4.4x 10xø 49 0.8 2.0x 10 -'• 18 x 10 5 wind. The point of this papertherefore is to exploreits conse- AR 471 Aug. 2 8.2 x 109 61 1.4 1.6 x 10 -3 25 x 105 74 1.8 1.5 x 10 -3 24 x 105 quencesin the solar wind and coronal holes. It is, however, AR 476 Aug. 11 1 x 10xø AR 488 Aug. 23 2 x 10xø 45 2.5 1.5 x 10-3 25 x 105 also interestingto note that (1) has been appliedrecently to BP 1 May 31 3.8 x 109 42 0.93 1.8 x 10 -3 28 x 105 severalproblems in astrophysics[Higdon, 1984, and unpub- BP2 May 31 2.5 x 109 37 0.5 1.1 x 10 -3 28 x 105 lishedreport, 1985]. We emphasize,however, that our work is BP3 Aug.21 6.3 x 109 59 1.5 2.1x 10 -3 27 x 105 not to be construedas a strongtest of MHD turbulenceprin- BP3' Aug.23 5.3 x 109 36 1.5 2.3 x 10 -3 30x 105 BP4 Aug. 21 3.5 x 109 30 0.87 1.8 x 10 -3 30x 105 ciplesin the solar wind. This is precludedboth by the signifi- LSS 2.4 x 10 xø 20 0.3 1.1 x 10 -'• 19x 105 cant uncertaintiesin our handling of the turbulenceand by the approximations used in the corona/wind model itself, as Data in the first five columns are from Golub et al. [1980]. The last will be discussed in the next section. two columnsare from (2) and (5), respectively.All units are cgs. 4116 HOLLWEG:TRANSITION REGION, CORONA, AND SOLARWIND

transition from (11) to (12). In generalthe transition was found This too is a lowest order approximation based on the fact to occur near 10 Rs. So we are here simply placing the transi- that pure Alfven waves propagate energy and information ex- tion at 10 Rs.) actly along the average field. In a turbulent situation, however, 5. The heat conduction flux is taken to be continuous at the Alfven wavesare not pure, and L .... will in generalevolve 10 Rs. This requirement servesas an outer boundary con- self-consistentlywith the turbulence. This aspect is ignored dition on the region inside 10 Rs. Strictly speaking,there is no here. boundary condition on the temperature at infinity; (12) The remaining equations are as follows. guaranteesthat T(r • oo) • O. Mass conservation 6. We use the radiative loss function for an optically thin coronal plasma. As summarized by Rosner et al. [1978], the d(pvA)/dr= 0 (20) volumetric energy lossrate is Momentum conservation Erad = ne2p(T•) (13) dv dp d (•B 2) GMs wherene is electronconcentration and P is approximatedby pvd-• = - dr dr 8re P 7- (21) P(T) • 10-21'2 104'9 < T < 10 TM whereG is Newton'sgravitational constant, Ms is solarmass, P(T) • 10-1ø'4T-2 105'4 < T < 105'?5 and p is pressure, (14) P(T) • 10-21'9 105'?5 < T < 10 6'3 p = 2pkT/me (22) P(T) .•, 10-17'?T -2/3 106'3 < T < 107 K (k is Boltzmann'sconstant, and m e theproton mass). Combin- (all units are cgs). ing (20)-(22) displaysthe critical point: 7. We presentonly a one-fluid model. Thus d/) /) = (15) dr l)2 -- 2R T whereT e denotesproton temperature. This is a goodapproxi- dT mation in the low corona, where Coulomb collisions are fre- [--GMs•5 8rcp1d(•B2> dr + 2R(r dlnA dr •l•')] (23) quent enough to equalize the temperatures.But it could be a poor approximation beyond r m 2 R s. A two-fluid model is whereR ---k/m r For a reason that will be discussedbelow, we also rewrite being developed. (21) and (22) in the form 8. The Alfven wave propagation is modeled via the follow- ing well-known [e.g., Hollweg, 1973, 1982] equation'

dS -- = -4r•(1 + M)Br- •Ea (16) dlndr p - 2RT1 (-- 2R dTdr GMsr 2 t9dvdr 8rcp1 d(gB2)) dr dr (24) where S is related to the wave action' Finally, the thermal energy equation is S --- (gB2)(1 + M)2(4rcp)-'/2 (17) dT dlnp 1 d ------(qA) + En- Lrad (25) and M is the Alfvenic Mach number of the flow' 3pRyc•r 2pRTv dr = Adr M = v(4rcp)•/2Br- • (18) Using (11) in r < 10 R s, this becomes Equation (16) expressesthe general energeticsof outward propagatingAllyen waves.Action is conservedin the absence A 2 --' +Lraa--En of irreversibleheating' if there is heating,then action is lost. It dlnp is correctfor any form of En, but it doesassume the validity of + 3pRvT'- 2pRy T (26) the small-wavelengthWKB approximationand equipartition dr whereK = KoT5/2. On theother hand, using (12) in r > 10Rs - (19) 2 6.25 i i i i Equation (19) is formally incompatible with a turbulent cas- cade, and thus (19) is only a lowest order approximation. Tu 6.00 et al. [1984'] have shownhow this lowestorder approximation can be developed to derive (16), but with En given by an • 5.75 appropriateturbulent cascade rate. For En we adopt (1). This differsfrom what Tu et al. derived,but we have alreadynoted i- 5.50 that (1) is alsocompatible with their development.Since En is given by (1) and (19), (16) can be construedas an equationfor o 5.25' 9. For the reason given above, the quantity L .... is as- sumed to scale as the mean distance between photospheric 5.00 I magnetic flux tubes, or equivalently, as the distance between 0.0 0.5 1.0 1.5 2.0 2.5 magnetic field lines. Thus Log (r/re) LcorrocB- 1/2 Fig.2. Temperatureversus heliocentric distance formodel 1. HOLLWEG:TRANSITION REGION, CORONA, AND SOLARWIND 4117

, - IO ' I ' I ' I ' Most of the points are close to the sun, in the transition region.

8 4. RESULTS As a referencemodel, we consider first the following bound- 6 ary conditions(model 1)' n(10Rs)= 5 x 103 cm-3 T(10 Rs)= 1.4 x 106K $(10Rs) = 6.834x 10'• Gcm s-• where n is proton (or electron)concentration. In addition, we 0 i I L I , I 0.0 0.2 havetaken L ..... s = 2.6 x 10'• km. Thismodel yields the fol- lowing conditionsat 1 AU'

Log(r/r e ) -3 n(1 AU)= 7.8 cm Fig. 3. Electron or proton concentration versus hclioccntric dis- tance.The solidcurve is from model 1. The dashedcurve is the polar T(1 AU)= 105K hole data of Munro andJackson [1977]. v(1 AU)= 536 km s-• nv(1AU)= 4.2 x 108cm-2 s-• yields The velocityis somehatlow and the densitysomewhat high, but otherwisethis modelis not badly representativeof a high- speedstream at 1 AU [W. Feldmanet al., 1976]. drd ln(3 ) F(En--Lraa)pv (27) The temperature profile is displayed in Figure2. Thereis a where broadtemperature maximum of 1.45x 106 K at r = 2.45 Rs. By comparison, the Parker critical point is at r = 2.01 Rs. Clearly,the solarwind flow cannotbe ignoredin calculating the temperatureprofile. The temperaturerises rapidly in the 3(1 + 0q2) transitionregion close to the sun,attaining 10 6 K at r = 1.1 This completesthe systemof equations.The method of Rs. This roughly agreeswith the observationalresults of Mariska [1978], who found T-- 106 K at r = 1.08 Rs for a solution is iterative. A velocity profile is guessed.With v(r) polarcoronal hole observed at the limb in August1973. (The given,the IMSL routineDVERK is usedto integrate(16), (20) discontinuousbehavior of T(r) at r- 10 Rs resultsfrom the or (24), and (26) inward from 10 Rs to the sun. These equa- artificiallydiscontinuous transition from classicalheat con- tionsrequire four boundaryconditions at 10 Rs.The valuesof duction to collisionlessheat conduction.All other quantities S, p, and T are specifiedat 10 Rs, and the fourth boundary in the model are well-behavedat r - 10 Rs.) condition comes from requiring continuity of q at 10 Rs Figure3 displaysn(r) in theregion Rs < r < 10Rs. The (points 4 and 5 above).The resultingupdated valuesof S(r), p(r), and T(r) are thenused to integrate(23) for v(r) inward rapiddecline of n closeto thesun coincides with the rapid rise and outward from the Parker critical point, which occurs at of T, and the total pressureis nearly constantin the thin that point in the v-r plane wherev 2 -2RT and wherethe transitionregion. At r = Rs Wehave nT = 9 x 10•3 (cgs), squarebracket in (23) for v(r) vanishes.This updatedvelocity whichis reasonablyrepresentative of conditionsat the baseof profile is then usedin a seconditeration. The iterations con- a low-pressurecoronal hole' for example,Mariska [1978] de- tinue until no quantities change by more than 2% between ducednT = 1.29x 10•'• (cgs)for a polar coronalhole at the successive iterations. limb,and Webband Davis [1985] give nT = 1.95x 10•'• (cgs) As noted above, a choice is made between (20) and (24). for low-pressureholes in 1974.However, Munro and Withbroe Equation (20) is usedwhen v2> R T; this ensuresaccurate preservationof the massflux. However,we have found by trial and error that (24) is more suitable close to the sun, where 7.25 .... -.25 v2 < RT and the corona is closeto hydrostaticequilibrium; simultaneousintegration of (24) and (26) will automatically yield the near constancyof pressureacross the thin transition region. However, in no caseis massconservation violated by 6.75 - -.75 more than 1% or so. One final point is the following: for each model, the values 6.50 of p and T at 10 Rs are pickedfirst. Thesevalues are generally chosento yield a reasonablemass flux at 1 AU. The value of S /////• % --I at 10 Rs is then repeatedlyguessed until a reasonabletransi- tion region develops.If S(10 Rs) is too small, then there are / 6.?_5// /// % --I.?-5 insufficient waves to heat the corona, and no transition region 6.00 ' ' I I , -I.5 is obtained. On the other hand, if S(10 Rs) is too large, the 0.0 0.5 1.0 1.5 2.0 2.5 transition region occurswell above the solar surface. Log(r/r e ) After the solutionhas convergedin r < 10 Rs, (16), (20), (23), and (27) are integratedoutward from 10 R s to 1 AU. Fig. 4. The quantities(6v•-) •/•- (solid curve) and The total number of points in the numerical grid is 1634. (dashedcurve) versus hclioccntric distance for model 1. 4118 HOLLWEG: TRANSITIONREGION, CORONA,AND SOLARWIND

0.0

-0.5

-2.0 Q,R

-2.5 ' ' ' ' I I I I 0.0 0.5 1.0 1.5 2.0 2.5 0 .01 .oa .04 .05 Log (r/r e ) Log (r/r e ) Fig. 5. The normalized wave "action" versus heliocentric distance Fig. 7. Same as Figure 6, but on an expanded scale to show the for model 1. behavior at the base of the transition region.

[1972] report nT = 3.16 x 10x4 (cgs)at the centerof a coro- and Feldman, 1977; Doschek et al., 1981; U. Feldman et al., nal hole, Withbroeand Noyes [197½] give nT = 2.5 x 10x4 1976]. The rms velocity reachesa maximum of 126 km s-x (cgs),Withbroe and Wang [1972] give nT = 3.8 x 10•4 (cgs), near r -- 2.8 Rs. The presenceof coronal velocity fluctuations Kopp and Orrall [1976] give nT =4.5 x 10x4 (cgs), and of 100-200 km s-x has beensuggested by a variety of radio Rosnerand Vaiana [1977] deducenT = 4.35 x 10•4 (cgs).The studiesof the corona, but these studiessuggest that the maxi- pressure in this model may therefore be too low; we will mum rms velocities occur near r = 10 R s [Armstron•l and return to this point below. Woo, 1981]. However, the radio observations are rather Also shown in Figure 3, by the dashed line, is the central scanty (especiallyclose to the sun) and subjectto a number of density in the polar coronal hole observed by Munro and uncertainties,so it is not clear if the discrepancyis real. If it is Jackson [1977] in July 1973. The model density profile is real, it may imply that the wavesare being dissipatedtoo close slightly flatter than the observed,but since observational error to the sun in the model. At 1 AU the model shows rms veloci- bars are not given, it is difficult to assesswhether the differ- ties of 14 km s-•. Velocity fluctuationsof this magnitudeor ence is significant. The model densities are higher than the larger have been associated with Alfven waves in the solar observedby about a factor of 2. This differencemay merely be wind [e.g., Belcherand Davis, 1971, Figure 1]. a consequenceof our having tried to obtain a model that Figure4 alsodisplays (cSB2)X/2/B versus r. Exceptnear the matches observed solar wind mass fluxes near the ecliptic, base of the transition region, this quantity increaseswith in- whereasthe observationsrefer to a polar flow; this notion is creasingr, attaining a value of 0.35 at 1 AU; this implies that consistentwith Figure 6 of Saito et al. [1977]. the waves are modestly nonlinear at 1 AU. Relative magnetic The behaviorsof the wave propertiesare displayedin Fig- field fluctuations of this magnitude or larger have been associ- ures4 and 5. Figure 4 displays(•¾2)1/2 versus r. The rms ated with Alfven waves at 1 AU [e.g., Bavassanoet al., 1982a; velocityfluctuation is 30 km s- x at the baseof the model.This Behannon, 1978; Belcher and Davis, 1971, Table 3; Denskat and would correspondto a line-of-sightrms velocityof 22 km s-x, Neubauer, 1982]. if the corona were observed at the limb. Velocity fluctuations The fact that the model yields both velocity and magnetic of this order are indeed observed in coronal holes [Doschek field fluctuations that are somewhat low at 1 AU may also

0 I i i i .•. 10-4 •, i i I ! _•' W T E

• io5 _ 0 -6 o, IO e Q x R

LL , o- -8 >, 104 - a) IO .c_

E 103 • W 0 .5 I 1'.5 2 2.5 0.00 0'01 o'.o o'.o, OD4' 0.05 LocJ(r/r e) Log(r/r o) Fig. 6. Energy fluxes,i.e., (flux density)x (A/Ao) versusheliocen- tric distance for model 1. The symbols are E (enthalpy flux), K (kinetic Fig. 8. The magnitudesof severalterms in (26) versusheliocentric energy flux), Q (magnitude of the heat conduction flux), G (magnitude distance for model 1. The curve denoted E representsthe enthalpy of the gravitational energy flux), W (wave energy flux) and, R (radi- terms, i.e., the last two terms on the right-hand side of (26). Note that ative energy flux). the enthalpy terms are important and sometimesdominant. HOLLWEG' TRANSITION REGION, CORONA, AND SOLAR WIND 4119

! I ! I ! I ! I ! I ] I ! I 6.25

10iø _ 6.00

T "-- -- 160 MHz 5.75

_ • '-• 80 MHz i-- 5.50

o ._j 5.25

5.00

4.75 i i I I ! I , I I I I I • I • / 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 .6 Log(r/r e) T (106K ) Fig. 10. Same as Figure 2 for model 2. Fig. 9. The differential emissionmeasure (dashed curve) and elec- tron concentration(solid curve) versustemperature for model 1. The cross-hatchedregion representsobserved differential emission mea- The heat flux reacheszero at the base of the model, where sures,and the horizontal lines representobserved values of n(T) as T = 2 x l0 s K. At thispoint the modelfails. The waveheat- given by Rosnerand Vaiana [1977]. ing is too smallto balancethe radiativeand enthalpylosses at lower temperatures,and the model fails to producetemper- aturesbelow 2 x l0 s K. The imbalancebetween heat input indicate that the wave dissipationoccurs too closeto the sun and radiative and enthalpylosses is illustratedin Figure 8. in the model. The curve marked E representsthe last two terms on the The rate at which the wavesdissipate is illustrated graphi- right-handside of (26); theseare the enthalpyterms. It is clear cally in Figure 5, which displays$/Ss versusr. Recall that $ that the radiative and enthalpy lossesoverwhelm the heat would be constantif the wavesdid not dissipate.The mono- input close to the sun. Figure 8 also illustratesthe general tonic declineof $ in Figure 5 is a result of dissipation'see (16). point that the enthalpy terms are important componentsof Note that more than two-thirds of the "action" is lost in the (26); this holds true in the presentcase all the way out to the first two solar radii above the surface of the sun. temperature maximum. The energybudget is displayedin Figure 6, which showsthe Another failure of the model is illustratedby Figure 9. Here energy fluxes (i.e., flux density multiplied by A/A s, where A is we displayn as a function of T in the transitionregion. Also the cross-sectionalarea of a stream tube) between the sun and shown, by the dashed curve, is the differential emission mea- 1 AU' seethe caption for an explanationof the symbols.The suren2/(dT/dr). Rosner and Vaiana[1977] havediscussed how heat conductionis negative(inward) closeto the sun, and the n(T) can be deduced from radio observationsof coronal holes. gravitational energy flux is always negative' the figure indi- Results at two frequenciesare displayed as the horizontal catestheir magnitudesonly. Virtually all of the energyis pro- bars, which are to be compared with the solid theoretical vided by the waves, which have an energy flux density of curve. Evidently, the model is underdense in the transition 4.5 x l0 s ergs cm-2 s- 1 at the sun. most of this flux either region. Rosner and Vaiana have also discussedhow the differ- goes into the kinetic energy flux of the distant solar wind or ential emissionmeasure can be deducedfrom X ray and EUV into the energyrequired to lift the wind out of the sun'sgravi- observations.The general trend of their observationalresults tational well. for a high-pressurecoronal hole is displayedby the cross- The energybalance below the temperaturemaximum is par- hatchedarea in Figure 9, which is to be comparedwith the ticularly interestingfrom the standpoint of coronal heating. dashedtheoretical curve. The most likely explanationfor the Note that the downward heat conduction flux at first increases discrepancyis that the model is increasinglyunderdense at with increasingdistance from the temperaturemaximum. This higher temperatures. occursbecause the plasma is heated,and the energythus de- positedis carried away by an increasingdownward heat flux. 7.00 ! i ! 'l ...... 6 The downward heat flux density has a maximum value of about3 x 10'• ergscm-2 s-• nearr • 1.2Rs. Downward heat 6.75 flux densitiesof this magnitude have been deducedby Mariska E [1978] for the coronal hole at the limb, but in his model the heat flux remains essentially constant down to much lower • 6.50' heights, r • 1.03 Rs. In contrast, the heat flux in our model drops precipitouslybelow r • 1.2 Rs. This occursbecause the >• 6.2:5 wave heating En is no longer able to balancethe radiative and // V enthalpy losses,and the deficit is taken out of the heat flux. o', 6 00, From Figures 6 and 7 it is apparent that the downward heat o flux is converted into enthalpy flux and into radiation; the 5.75 -I.6 radiativeflux appearingin Figures6 and 7 is 0.0 0.5 1.0 1.5 2.0 2.5

R -- LradA(r)/As dr (28) Log(r/r e) s Fig. 11. Same as Figure 4 for model 2. 4120 HOLLWEG:TRANSITION REGION, CORONA, AND SOLAR WIND

i i i i i i 106 ! ! ! 10I• _

i T I •160 MHz • 105 i

x

>, 104 106

103 104, , , , , 0 .5 1.0 1.5 2.0 2.5 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Log(r/r e ) T (106K ) Fig. 12. Sameas Figure 6 for model 2. Fig. 14. Same as Figure 9 for model 2.

Thuswe havea modelthat containsthe correctqualitative To showthe effect of a smallerchoice for L½.... $, we present featuresof a transitionregion, temperature maximum, and now a modelwith the followingboundary conditions (model high-speedsolar wind flow. But the model fails in detail. It is 2)' simultaneouslyunderdense in thetransition region and some- what overdenseat 1 AU. The transitionregion pressure is too n(10Rs)= 5 x 103cm -3 low.From the standprint of coronal heating, the model seems T(10 Rs)= 1.4 x 106 K to requiremore heating closer to the sun.This wouldhelp the modelin two ways'(1) increasingEa woulddirectly help to S(10 Rs) = 2.072 x 10'• G cm s-1 balancethe radiative and enthalpylosses in the transition region, and (2) depositingmore heat closerto the sun would L½.... $ = 1.3 x 10'• km increasethe coronal iemperature and lower the height of the The model yieldsthe following conditionsat 1 AU' temperaturemaximum; the resultingincrease of dT/dr and T,•axwould increasethe downwardheat conductionflux, and n(1 AU)= 7.7 cm-3 thiscan then help to balancethe radiative and enthalpy losses T(1AU)=8.8 x 10'•K in the transitionregion. However, increasing dT/dr alsoin- creasesthe enthalpy losses in thetransition region; see (26). v(1 AU)= 483 km s-1 Within the context of our presentmodel, an increasein heatingclose to thesun can be effected by choosinga smaller nv(1AU)= 3.7 x 108 cm-2 s-x valuefor L½.... s-As we shallnow see,this doesimprove the Comparingwith the previousmodel, we seethat the velocity situationin the transitionregion but only in somerespects. and temperatureat 1 AU have beenreduced by 10% and But it simultaneouslyworsens the behaviorof the model vis-a- 12%, respectively.This occurs becausethe increaseddissi- vis the distant solar wind. The reason is that a smaller choice pationclose to thesun leaves less wave energy for heatingand for L½.... s increasesthe wavedissipation close to the sun and acceleratingthe distant solar wind. thusleaves less wave energy for acceleratingand heatingthe The temperatureprofile is displayedin Figure10. The tem- distant wind; recall that in the previousmodel, the wave am- peraturemaximum is now at r = 1.88 Rs, and Tmax= 1.52 plitudes(•B 2) x/2and (•v 2) x/2were already on the low sides x 106 K. ReducingLcorr increases Tma xand moves the temper- of the observedvalues at 1 AU, and the maximumof (•v 2) occurred too close to the sun. aturemaximum closer to the sun.The Parker criticalpoint is now at r = 2.09 Rs; we again find that the temperaturemaxi- mum is not far from the criticalpoint, and self-consistency with the solarwind flow is an essentialaspect of the coronal •- 10-4k ' ' ' ' ' heating. 'u• EH The densityprofile is similarto that shownin Figure3. The - only exceptionis veryclose to the sun,where the highertem- o iC•6 _ peraturesin the presentmodel result in a flatter densitypro- file. Atr-R s we haven--2.26x 108cm-3, T=2x l0 s K, and nT =4.5 x 10•3 (cgs).The pressurein the transition region is now distinctlylower than has been observedin coro- (u 10-8 nal hole regions. .c::_ The behaviorsof the wave propertiesare displayedin Figure11. Comparingwith Figure4 showsthat (•v 2) x/2and E (cSB•)X/•/Bare markedly reduced at 1 AU. Thisagain illus- , i i , / tratesthe point that increasingthe dissipationnear the sun 0.00 0.01 0.02 0.0:5 0.04 0.05 leavestoo little waveenergy at 1 AU. Note too that the maxi- Log(r/r e ) mumof (•v :) 1/:is now closer to thesun, worsening the possi- ble disagreementwith the radio observations. Fig. 13. Same as Figure 8 for model 2. The energybudget is displayedin Figure12. Virtually all of HOLLWEG:TRANSITION REGION, CORONA, AND SOLAR WIND 4121

6.5 7.0 .... -I.2

6.0 E 6.5 -I.3 __ • /// \ '-' 5.5 6.0 I- -I.4 5.0 5.5 - / o / / / 5.0 • I I I I -I.5 4.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 1.5 2:.0 2.5 Log(r/r e) Log(r/r e ) Fig. 17. Sameas Figure 4 for model 3. Fig. 15. Same as Figure 2 for model 3.

the energy is provided by the waves, which have an energy creasing temperatures than was the case in the first model flux densityat the sunof 4 x 105ergs cm-2 s -•. Comparison (Figure 9). This is becausethe temperature maximum has with Figures 6 and 7 shows a significantchange in the down- moved closer to the sun. ward heat flux near the sun. The maximum downward heat A model with higher pressurein the transition region can be flux densityhas increasedto 6 x 104 ergscm-2 s-•; this is obtainedby changedn and T at 10 Rs (model 3): very nearly the value deduced by Mariska [1978] from limb observations of a coronal hole. Moreover, the downward heat n(10Rs)= 104cm -3 flux is nearly constant below r • 1.1 Rs; this too agreeswith T(10Rs)=8 x 105K Mariska's results. There is some loss of heat flux below r • 1.02 Rs, which goesinto enthalpy and radiation. In con- S(10 Rs)= 2282.2 Gcm s- • trast to the previous model, the downward heat flux does not L ..... s - 6000 km vanishat the base,and temperaturesbelow 2 x 105K can be attained; however, the temperature gradient has become very The model yieldsthe following conditionsat 1 AU: large at this temperature, and we have not continued the inte- gration for reasonsof numerical accuracy. n(1 AU) = 13.2 cm- 3 Of course, the improved behavior of the downward heat T(1 AU)=4.3 x 104K flux is only partially due to increasingTruax and moving the temperature maximum closer to the sun, thereby increasing v(1 AU)= 271 km s- • dT/dr. It is due also to the lower radiative lossesin the lower nv(1 AU)= 3.57 x 108 cm- 2 s- • pressuretransition region. Comparing Figure 12 with Figures 6 and 7 reveals that the radiative losseshave been reduced by In this model,the pressureat the baseof the transitionregion almost an order of magnitude. The reduction of the radiative is nT = 2.15 x 10TM (cgs). The higherpressure implies larger lossesis also evident when Figure 13 is compared with Figure radiativelosses. To balancethese losses, the heatingin the low 8. coronamust be increasedby reducingL .....$. As we discovered The differential emission measure and n(T) curves for this above, the enhanced coronal dissipation leaves less wave model are displayed in Figure 14, where it is once again ap- energy for heating and acceleratingthe distant solar wind. parent that the model is underdensein the transition region. Consequently,this model only yields an unrealisticallylow- Note, however,that both curvesfall off less rapidly with in- speedflow at 1 AU. The temperatureprofile for this caseis displayedin Figure 15. Note that the temperaturemaximum is now at r = 1.3 Rs, 10 which is closerto the sun than in the previoustwo examples. By comparison,the Parker critical point is now at r = 6.18 R•; in this case the temperature maximum is well inside the criti- cal point.The maximumtemperature is Tmax= 1.33 x 10a K.

• 6 The coronal densityprofile is displayedin Figure 16, along 'E with the polar hole density measuredby Munro and Jackson [1977]. With the exceptionof the steepdecline near r = 2 Rs, ,.-- 4 the observedand calculatedslopes agree very well. As before,

o the difference in absolute values may be merely due to our • 2 comparing polar data with a model that tries to match ob- servedmass fluxes in the ecliptic. 0 I I i I i I The behaviorsof <•¾2)1/2and <6B2)•/2/Bare displayedin 0.0 0.2 O.4 O.6 0.8 .0 Figure 17. In this model the wave dissipationclose to the sun is so severethat only an unrealisticallysmall flux of wave Log (r/re) energysurvives at 1 AU. Fig. 16. Same as Figure 3 for model 3. The energybudget betweenthe sun and 10 Rs is displayed 4122 HOLLWEG' TRANSITIONREGION, CORONA, AND SOLARWIND

iO6 -4 T(n ' , ' EH

CO - L o,J8 ._

io3 o .2 .4 .6 .8 ß o o.ol' o'.oz o'.o 0.04 o.05 Log (r/r e) Log (r/r e) Fig. 18. Same as Figure 6 for model 3. Fig. 20. Same as Figure 8 for model 3. Note that the enthalpy terms are lessimportant in this case.

in Figure 18. The waveenergy flux densityis 3.3 x 105ergs Finally, the differentialemission measure and n(T) are dis- cm-2 s-• at the sun. Note the rapid declineof the wave played in Figure 21. The n(T) curve passeswithin the error energy flux, which is a consequenceof the small value of bars on the radio data. The differential emission measure is Loorr,S.In this model the wave energygoes mainly into the uniformly about a factor of 2 below the observedvalues. This kinetic energy of the distant solar wind, into the energy re- is primarily a result of our having chosena transitionregion quired to lift the solar wind out of the sun'sgravitational well, pressurethat is a compromisebetween the values given by and into radiation; this is to be contrastedwith the previous Mariska [1978] and by Rosnerand Vaiana [1977]. two models,in which radiation was not an important part of the overall energy budget. The downward heat conduction 5. DISCUSSION flux increasesstrongly below the temperature maximum. This is a consequenceof the strong wave heating in that region. Our previous wave-driven solar wind models [Hollweg, The maximum downward heat conductionflux densityis close 1978] have been extended by including a new hypothesisfor to l0 s ergscm-2 S-1, occurringat r • 1.014Rs. This is some- the nonlinear wave dissipation. The hypothesis is that the what larger than the value deduced by Mariska [1978] but wavesdissipate via a turbulent cascadeat the rate given by (1) somewhat smaller than the value appearing in the model and that the waves evolve accordingto (16). A subhypothesis (model 2) of Rosner and Vaiana [1977]. The differencesare is that the relevant correlation length scales as the distance due largely to the different transition region pressures(and betweenmagnetic field lines, so that L .... OCB- •/2. therefore different radiative losses)in the various models: in This hypothesislacks firm theoreticalsupport, but Tu et al. Mariska's model, nT = 1.29 x 10• '• (cgs); in Rosner and [1984] have indicated how a formal developmentcould lead Vaiana,nT = 4.35 x 10•'• (cgs),while nT = 2.15 x 10•'• (cgs) to (1) and (16). We have also provided empirical evidence in the presentmodel. showing that (1) is consistentwith observationsof the solar The behavior of the energybudget in the transition region is chromosphere,solar coronal active region loops, and the dis- displayed in Figure 19. The downward heat flux decreases tant solar wind. To some extent, the present paper can be with decreasingr near the base of the model. This is because regarded as an attempt to find out whether our hypothesisis the wave heating cannot balance the radiative and enthalpy also consistent with what is known about the transition losses there, and the deficit is taken out of the heat flux. This region,corona, and solar wind in coronal holes.In particular, can also be seen in Figure 20, which displaysthe magnitudes we sought to learn whether it was possibleto produce a wave- of the various terms on the right-hand sideof (26). driven solar wind model that puts the corona and transition region on the same footing as the solar wind itselfi Previous wave-driven models have emphasized the importance of heat- ing in the distant solar wind, but they have completely ignored 106 the coronal heating, and instead assumed a hot corona as a

W boundary condition. We have succeededin producing models that exhibit the correct qualitative features,viz., a steep temperature rise to a maximumcoronal temperature in excessof 106 K and a sub- stantial solar wind mass flux in excess of 3.5 x 108 cm-2 s- • at 1 A U. Another favorable feature of the model is its agree- ment with observed nonthermal velocities at the bases of coro- nal holes. However, the model fails in detail. In order to produce a high-speedsolar wind stream,it was necessaryfor a substan- iO3 tial fraction of the available wave energyto propagateinto the ß I .o2i .o3i .o4i .05 supersonicportion of the solar wind. This in turn required a Log(r/r e ) relativelylong dissipationlength, i.e., large valuesof Loorr.As a consequence,the corona and transition region were under- Fig. 19. Same as Figure 7 for model 3. heated,and the modelsyielded basepressures that were lower HOLLWEG' TRANSITIONREGION, CORONA, AND SOLARWIND 4123

approximation,which gives ((•¾2)1/20C10 -1/4' for undamped waves in a stationary or slowly moving plasma; note that p is ioIo a rapidly increasing function of depth near the base. Thus En scalesas p•/4 for weaklydamped WKB waves.On the other hand, waveswith periods longer than a hundred secondsor so T have wavelengths that are long compared with the density scale height near the base of the model. Those waves should see the transition region and low corona essentially as a dis- continuity,across which <1•V2>1/2• const. In that case,En wouldscale as p, insteadof pI/4. Thusour WKB analysismay r- 106 _ have underestimatedthe transition region and lower coronal '0 ///• heating. Unfortunately, a non-WKB analysis is well beyond the scopeof this paper; there is no simple analog to (16), and a 104 I • I • I , •'/I • I , full wave analysiswould be required. 0.0 0.2: 0.4 O.G 0.8 1.0 1.2 1.4 Our model also contains other uncertainties that might T(106K) make the complete abandonment of the hypothesizedequa- tion (1) premature.In particular are the following. Fig. 21. Sameas Figure 9 for model 3. 1. Our treatment of qe is really a guess.As mentioned in the introduction, our use of the classicalformula inside 10 Rs than observedpressures. On the other hand, if L .... was ad- could well be in error. Heat conductionplays an important justed downward to give reasonablebase pressures,then not role in the transition region, and that is where our high-speed enough wave energy could penetrate into the solar wind stream model runs into difficulty. However, we note that beyond the critical point, and only a low-speed flow could be Shoub [1982] has considerednonclassical modifications to qe produced. We have been unable to find a set of parameters in the transition region and found deviations by no more than that simultaneouslyyields a high-speedsolar wind flow and a 40% from the classical value. reasonablebase pressure. 2. Our modelis one-fluid,with T• = Tv. The effectsof a Does this then mean that the hypothesisembodied in (1) two-fluid model are difficult to gauge,especially in the absence and (16) should be abandoned?This has to be admitted as a of a theory for the dissipation range of the turbulence. Such a real possibility.On the other hand, the model "comesclose." theory would be required to properly apportion En between In the first model presentedin section4, a reasonablehigh- the electronsand protons. speedsolar wind stream was obtained,but the base pressure 3. Our results indicate the sensitivity of the model to the was too low. On the other hand, the base pressure was not relative amounts of wave energy available inside and outside disastrouslylow; we obtainednT = 9 x 10•3 (cgs)compared the critical point. Thus the results should be sensitiveto the with Mariska's value nT = 1.29 x 10•'• (cgs).So beforecom- location of the critical point. The models in section 4 verify pletely abandoning our model, we are inclined to ask whether this claim indirectly. In the first two models the critical point the model can be improved by relaxing some of the sim- was at r = 2.01 Rs and r = 2.09 Rs, respectively.With the plifyingassumptions that wereemployed. critical point so close to the sun, it was reasonablyeasy for Let us supposethat there is in fact sufficientwave energy wave energy to be deposited in the solar wind beyond the flux available beyond the critical point to heat and accelerate critical point, with the result that reasonablevalues for v and a high-speedsolar wind stream,as in the first model in section T were obtained at 1 AU. In the third model, however, the 4. For that energy flux, an improved model would require critical point was at r = 6.18 Rs. The increasedcritical radius larger values of E n in the transition region and low corona; made it even more difficult for wave energyto be depositedin this is required in order to balance the radiative lossesfrom a the wind, and an unacceptablylow flow speedwas obtained at higher pressurehole. Two ways to accomplishthis are as fol- 1 AU. Thus in a situation with multiple X-type critical points lows. [Kopp and Holzer, 1976; Holzer, 1977], the solution passing 1. The heating can be increasedby reducingL .... in the through the innermostcritical point will be the most favorable transition region and low corona relative to its values beyond for producinga high-speedwind. the critical point. This can be accomplishedby increasingfmax. 4. There are uncertainties concerning the observational It can also be accomplishedby allowingfor a rapid expansion data that the model is trying to reproduce. Further observa- of the magnetic flux tubes with height, associated with the tions of "turbulent" velocities in the corona and inner solar network. The network expansionwas emphasizedby Gabriel wind (as by Armstrong and Woo [1981] and by G. L. [1976] and by Rosner and Vaiana [1977]; the latter authors Withbroe et al. (unpublished report, 1985) are mandatory. restricted "the extra expansion to the temperature range 7 Coronal magnetic field measurementsare painfully elusive; x 105 K < T < 106 K," and took "a total fractional increase Faraday rotation data [Hollweg et al., 1982] may be the best in cross-sectional area of the order of 2-3, since the network possibility. Finally, there are observational uncertaintiescon- occupies approximately 40% of the total area at cerning the transition region. As Mariska [1978] has empha- chromosphericheights." sized [see also Mariska and Hollweg, 1985], the solar atmo- 2. The WKB approximationbreaks down closeto the sun. sphere below 20,000 km is manifestly inhomogeneousand We have already mentioned that this probably does not probably dynamic.It is thereforenot clear whetherit is mean- seriouslyaffect the calculationof the wave pressure,which is ingful to compare data below 20,000 km with a simple steady not important close to the sun. But the breakdown of WKB one-dimensionalmodel such as has been presentedhere. may seriouslyaffect the calculation of En via its sensitivede- 5. Finally, (1) is an empirically motivated Ansatz. This pendenceon (•v2) 1/2.From Figures3, 10, and 16 we note fact, combined with the other above-mentioned uncertainties, that (6v2) •/2 is a rapidlydecreasing function of depthnear the means that our model cannot be construedas a strong test of base of the model. This result is largely due to the WKB the applicabilityof turbulenceconcepts to the solar wind. 4124 HOLLWEG: TRANSITIONREGION, CORONA,AND SOLARWIND

Acknowled•lments.The author gratefullyacknowledges the valu- Doschek, G. A., J. T. Mariska, and U. Feldman, Mass motions in able assistanceof C. J. Pollock, who developedthe numericalcodes optically thin solar transition zone lines,Mon. Not. R. Astron.Soc., usedin an earlier versionof this work. He also acknowledgesuseful 195, 107, 1981. conversationswith A. Barnes, M. Dobrowolny, J. Higdon, W. Mat- Durney, B. R., Solar wind properties at the earth as predicted by thaeus, S. Olbert, C. Smith, C.-Y. Tu, and G. Withbroe. This work one-fluid models, J. Geophys.Res., 77, 4042, 1972. was supportedby the NASA Solar-TerrestrialTheory Program(grant Durney, B. R. Solar wind properties at earth as predicted by the NAGW-76) and by NASA grant NSG-7411. two-fluid model, Sol. Phys., 30, 223, 1973. The Editor thanks W. Matthaeus for his assistancein evaluating Durney, B. R., and A. J. 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