Extended MHD modeling of the steady solar corona and the solar wind
Tamas I. Gombosi, Bart van der Holst, Ward B. Manchester, Igor V. Sokolov
LLiving Rev. Sol. Phys. manuscript No. (will be inserted by the editor)
Extended MHD modeling of the steady solar coronaand the solar wind
Tamas I. Gombosi · Bart van der Holst · Ward B. Manchester · Igor V. Sokolov
Received: date / Accepted: date
Abstract
The history and present state of large-scale magnetohydrodynamic(MHD) modeling of the solar corona and the solar wind with steady or quasi-steady coronal physics is reviewed. We put the evolution of ideas leading to therecognition of the existence of an expanding solar atmosphere into historical con-text. The development and main features of the first generation of global coronaand solar wind models are described in detail. This historical perspective is alsoapplied to the present suite of global corona and solar wind models. We discuss theevolution of new ideas and their implementation into numerical simulation codes.We point out the scientific and computational challenges facing these models anddiscuss the ways various groups tried to overcome these challenges. Next, we dis-cuss the latest, state-of-the art models and point to the expected next steps inmodeling the corona and the interplanetary medium.
Keywords
Sun · Solar wind · MHD · Global simulations · Space weather
T. I. GombosiCenter for Space Environment Modeling,University of Michigan2455 Hayward, Ann Arbor, MI 48109, USAE-mail: [email protected]. van der HolstCSEM, University of MichiganE-mail: [email protected]. B. Manchester IVCSEM, University of MichiganE-mail: [email protected]. V. SokolovCSEM, University of MichiganE-mail: [email protected] a r X i v : . [ a s t r o - ph . S R ] J u l Tamas I. Gombosi et al.
Contents
It was realized thousands of years ago that the space between heavenly objectsmust be filled by something that is much lighter than the materials found onEarth.
Aether (ancient Greek for light ) was one of the primordial deities in Greekmythology. He was the personification of the upper air. Later the name “aether”was used by Greek philosophers to describe a very light fifth element. Plato men-tioned that “there is the most translucent kind which is called by the name ofaether.” Aristotle introduced a new “first” element to the system of the classi-cal elements (Earth, Fire, Air, Water). He noted that the four terrestrial classicalelements were subject to change and naturally moved linearly. The first elementhowever, located in the celestial regions and heavenly bodies, moved circularlyand had none of the qualities the terrestrial classical elements had. It was neitherhot nor cold, neither wet nor dry. With this addition the system of elements wasextended to five and later commentators started referring to the new first one asthe fifth and also called it aether. Medieval scholastic philosophers granted aetherchanges of density, in which the bodies of the planets were considered to be moredense than the medium which filled the rest of the universe. Later, scientists spec-ulated about the existence of aether to explain light and gravity. Isaac Newtonalso suggested the existence of an aether (Newton 1718).The idea that the Sun might be the source of corpuscular radiation was firstsuggested by Carrington (1860). On September 1, 1859, Carrington (1860) andHodgson (1860) independently observed a huge white-light solar flare. Less thana day later telegraph communications were severely disrupted during a planetary-scale magnetic storm. At the same time a great aurora was seen, even in Rome,a truly exceptional event (Green and Boardsen 2006). On September 2, the tele-graph line between Boston and Portland (Maine) operated on “celestial power,”without batteries (Loomis 1860). Carrington (1860) suspected a causal relation-ship between the solar flare, the magnetic storm, and the aurora, and he suggesteda continuous stream of solar particles as a way to connect these phenomena.Carrington’s suspicion, however, was not universally shared. Thirty-three yearsafter the so-called “Carrington event”, Lord Kelvin (William Thomson), in a Pres-idential Address to the Royal Society, argued that the Sun was incapable of power-ing even a moderate-sized magnetic storm: His argument was based on his unwill-ingness to think outside the box and his not accepting the possibility that the Sunmight be powered by a process that was not understandable in terms of “classical”physics. He was trying to explain the Sun’s energy production within the frame-work of coal burning. His logic lead him to the conclusion that the Sun could notbe older than a million years. Rather than questioning his own basic assumptions,he questioned Darwin’s estimates that some fossils might be hundreds of millionsof years old. At the end of his talk he confidently concluded, “It seems as if we mayalso be forced to conclude that the supposed connection between magnetic stormsand sunspots is unreal, and that the seeming agreement between the periods hasbeen a mere coincidence” (Thomson 1893). This reminds us of the old quote from
Michel de Montaigne: “Nothing is so firmly believed as what we least know.”The idea of a charged corpuscular radiation emanating from the Sun was nextsuggested by FitzGerald (1892) who wrote, “a sunspot is a source from which someemanation like a comet’s tail is projected from the Sun ... . Is it possible, then,that matter starting from the Sun with the explosive velocities we know possible
Tamas I. Gombosi et al. there, and subject to an acceleration of several times solar gravitation, could reachthe Earth in a couple of days?”A few years later, R¨ontgen (1896) discovered that cathode rays could causecrystals to fluoresce, pass through solid objects, and affect photographic plates.A few years later, Lodge (1900) suggested that magnetic storms were due to “atorrent or flying cloud of charged atoms or ions”; that auroras were caused by “thecathode ray constituents ... as they graze past the polar regions”; that comet tailscould not be accounted for by solar electromagnetic radiation pressure but couldbe accounted for by particle radiation emanating from sunspots “like a comet’stail” and “projected from the Sun” with an “average velocity [of] about 300 milesper second”; and finally, that “there seems to be some evidence from auroras andmagnetic storms that the Earth has a minute tail like that of a comet directedaway from the Sun” (quotes selected by Dessler 1967).The electron was discovered at the very end of the 19th century (Thomson1897) and the concept of an electrically neutral solar radiation composed of oppo-sitely charged particles was not introduced until the middle of the 1910s (Birkeland1916). He wrote: “From a physical point of view it is most probable that solar raysare neither exclusively negative nor positive rays, but of both kinds.”From his geomagnetic surveys, Birkeland (1908) realized that auroral activitywas nearly uninterrupted, and he concluded that the Earth was being continu-ally bombarded by “rays of electric corpuscles emitted by the Sun.” His workwas, however, pretty much ignored at the time. Some seventy years later, Dessler(1984) wrote an intriguing article arguing that personality conflicts between Sid-ney Chapman and Scandinavian scientists, as well as British imperial arrogance,were partly responsible for ignoring the pioneering works of Birkeland, Enskog,Størmer and Alfv´en.In the early 1930s, Chapman and Ferraro (1931a,b, 1932a,b, 1933) publishedthe first quantitative model of an infinitely conducting quasi-neutral plasma beamand its interaction with a magnetic dipole. Chapman was fundamentally a math-ematician and he always tried to simplify physics phenomena to treatable mathe-matical problems. Ferraro was a PhD student looking for a thesis topic. Akasofu(1995) quotes Chapman on why he did not consider a continuous plasma streamfrom the Sun: “He [Lindeman] said it [a stream of gas] must consist of charges ofopposite signs in practically equal numbers, so that it could hold together. Linde-mann never tried to develop what would be the consequences on the Earth of theimpact of such a stream of gas. I made an attempt at that while I was Professorat Manchester in 1919–1924, but unfortunately I started at the wrong end; I tried to find out what would be the steady state, as if the stream had been going onforever. It didn’t work out; so I was still wanting to find out what would happen,and this was the subject I proposed for Ferraro.” In plain English, Chapman didnot consider the case of a continuous solar wind because he could not solve thisproblem mathematically. xtended MHD modeling of the steady solar corona and the solar wind 5 ρ d , the anti-sunward radiation pressure, F rad is inversely proportional to the square of the heliocentric distance, d h , and pro-portional to the cross section of the dust grain, πa ( a is the equivalent radius ofa spherical dust grain), F rad ∝ a /d h . The sunward pointing gravitational forceis proportional to the particle mass (4 πρ d a /
3) and inversely proportional to theheliocentric distance: F grav ∝ a /d h . Since the two forces point in the oppositedirection, have the same heliocentric dependence, but exhibit different dependenceon the grain size ( F rad ∝ a and F grav ∝ a ), the resulting effect is a complex“dust mass spectrometer” where each particle size is moving under the influenceof its own “reduced solar gravity.” This effect results in the broad and curved dusttail. Fig. 1
Comet Hale-Bopp (1997) showing two distinct tails – a broad dust tail (white) and anarrow ion tail (blue). (Source:
Tamas I. Gombosi et al.
For active comets the straight, narrow plasma tails (also called Type I tails) are10 –10 km long and, within a few degrees, always point away from the Sun. Ob-servation of various tail structures as they moved down the Type I tail determinedthat the acceleration in plasma tails ranged from about 30 to 300 cm s − andoccasionally even larger. This value was some three orders of magnitude largerthan any acceleration due to solar radiation pressure. Some major process wasmissing. In order to account for the observed large acceleration Biermann (1951)postulated the existence of a continuous “solar corpuscular radiation” composedof electrons and ions. Assuming that the antisunward acceleration of small irregu-larities in Type I comet tails was due to Coulomb collisions between electrons in aradially outward plasma flow from the Sun and newly ionized cometary particles,Biermann (1951) inferred a solar wind density and velocity of n sw ≈ ,
000 cm − and u sw ≈ ,
000 km/s that represents a particle flux ∼
500 times larger than waslater observed.The important consequence of Biermann’s (1951) idea was that if solar cor-puscular radiation is responsible for the antisolar acceleration of comet tails thenthe Sun evidently emits solar corpuscular radiation in all directions at all times.This follows from the fact that comets “fill” the heliosphere; antisunward pointingcomet tails were observed over the poles of the Sun as well as at low heliographiclatitudes. In addition, comets come by as frequently at sunspot minima as atsunspot maxima. Yet none fail to show an antisolar Type I tail. This means thatinterplanetary space must be completely filled with solar corpuscular radiation.Bierman’s idea was pretty much ignored by the solar physics community. Thefact that the Sun has a million-degree corona was first discovered by Grotian(1939) and Edl´en (1941) by identifying the coronal lines as transitions from low-lying metastable levels of the ground configuration of highly ionized metals (thegreen FeXIV line at 530.3 nm, but also the red line FeX at 637.4 nm). In the mid-1950s Chapman (1957) calculated the properties of a gas at such a temperature anddetermined it was such a superb conductor of heat that it must extend way out intospace, beyond the orbit of Earth. However, Chapman – and others – considered astatic corona that was gravitationally bound by the Sun. If the Earth is movingat about 30 km/s along its orbit around the Sun, the interaction between theEarth and the stationary solar corona could only result in very minor geomagneticdisturbances. Large geomagnetic storms, however, were already well observed atthat time.2.2 Solar windSometime in 1957, Ludwig Biermann visited John Simpson’s group at the Uni-versity of Chicago. While in Chicago he had extensive discussions with one ofSimpson’s postdocs who was working on the problem of cosmic ray modulationin the solar system. Biermann explained his comet tail idea to the postdoc, Eu-gene Parker. According his own recollection, Parker started to think about the
Biermann–Chapman puzzle: how to reconcile Chapman’s (1957) hot, highly con-ducting corona with Biermann’s (1951) idea of a continuously outward streamingfast solar corpuscular radiation. Parker’s solution to the Biermann–Chapman puz-zle was the idea of the continuous expansion of the hot solar corona, the solar wind(Parker 1958). xtended MHD modeling of the steady solar corona and the solar wind 7
In his seminal paper, Parker (1958) pointed out that the static corona has afinite pressure at infinity that exceeds the pressure of the interstellar medium bya large factor. He concluded that the solar corona cannot be static (however, asit was later pointed out by Velli (1994) and Del Zanna et al (1998), the situationis quite complicated). Next, he considered a steady-state spherically symmetricisothermal corona that expands with a velocity v ( r ), where r is the heliocentricdistance. In addition, he assumed that the electron and ion temperatures wereidentical and approximated the plasma pressure by p = 2 nkT , where n ( r ) is theion number density, T is the ion temperature, and k is the Boltzmann constant.Using the conservation of particle flux and the momentum equation, he obtainedthe following analytic relation for the plasma flow velocity: (cid:20) v v m − ln (cid:18) v v m (cid:19)(cid:21) = 4 ln (cid:16) ra (cid:17) + (cid:18) v esc v m (cid:19) (cid:16) ar (cid:17) − (cid:18) v esc v m (cid:19) − , (1)where a is the radius of the hot coronal base (Parker used a value of a = 10 km), v esc is the escape velocity at radius a ( v esc = 2 GM (cid:12) /a , where G is the gravitationalconstant and M (cid:12) is the solar mass), while v m is the most probable velocity ( v m =2 kT /m p , where m p is the proton mass). In Eq. (1), the integration constant waschosen to ensure that the velocity is real and positive for all r > a values. Inaddition, only solutions with v (cid:28) kT /m and v ( r → ∞ ) > v m solutions wereconsidered, since the expansion velocity at the base of the hot corona, v wasassumed to be very small and Parker was interested in a hydrodynamic escapesolution. This solution was later called the “solar wind.” Fig. (2) shows the classof escaping corona solutions (Parker 1958). Fig. 2
Spherically symmetric hydrodynamic expansion velocity v ( r ) of an isothermal solarcorona with temperature T plotted as a function of r/a , where a is the radius of the base ofthe solar corona and has been taken to be 10 km (Parker 1958). Tamas I. Gombosi et al. It is interesting to note that in Parker’s (1958) solution the expansion velocityreaches the most probable particle velocity at the heliocentric distance of r c = av esc / v m . Since it is assumed that the outflow velocity is small at the coronalbase, this means that hydrodynamic outflow can only take place if the v m > v esc / /r in interplanetary space. Themagnetic field lines will follow an Archimedean spiral described by rR s − ln (cid:18) rR s (cid:19) = 1 + v s R s Ω (cid:12) ( φ − φ s ) , (2) where Ω (cid:12) is the angular velocity of the Sun, φ is the solar azimuth angle and φ s is the azimuth angle of the magnetic field line at the distance R s (Parker(1958) assumed a value of R s = 5 R (cid:12) ). Using this simple model Parker (1958)also expressed the magnetic field vector at an arbitrary point outside the r = R s sphere: B r ( r, θ, φ ) = B ( θ, φ s ) (cid:18) R s r (cid:19) B θ ( r, θ, φ ) = 0 B φ ( r, θ, φ ) = B ( θ, φ s ) Ω (cid:12) ( r − R s ) sin θv s (cid:18) R s r (cid:19) , (3)where v s is the asymptotic speed of the solar wind that is reached at a heliocentricdistance, where B ( θ, φ s ) is the radial component of the magnetic field at the originof the field line (on the r = R s sphere).At large heliocentric distances (where r (cid:29) R s ) the radial magnetic field com-ponent decreases as B r ∝ /r , while the azimuthal component only decreases as B φ ∝ /r . This means that in the interplanetary space the magnetic field linesbecome more and more “wound up” as one moves further and further away fromthe Sun. This heliospheric magnetic field configuration is now referred to as the“Parker spiral.” A schematic of the interplanetary magnetic field line topology isshown in Fig. 3 (Parker 1958). Another important consequence of Parker’s (1958) solution was that, due tothe conservation of mass flux, it predicted that the particle density in the inter-planetary medium would decrease as n ∝ /r , much slower than the exponentialdecrease obtained in the case of a hydrostatic atmosphere. This result was con-sistent with Biermann’s (1951) conclusions, but it was very different than theaccepted model predictions. xtended MHD modeling of the steady solar corona and the solar wind 9 Fig. 3
Solar equatorial interplanetary magnetic field lines carried by outward-streamingplasma with velocity 10 km/sec (Parker 1958). Opposition to Parker’s (1958) hypothesis on the solar wind was immediateand strong. He submitted the manuscript to the
Astrophysical Journal in earlyJanuary of 1958. Here is how Parker remembers the events (Parker 2001):“... sometime in the late Spring of 1958 the referee’s report on the original paperappeared, with the suggestion that the author familiarize himself with the subjectbefore attempting to write a scientific article on solar corpuscular radiation. Therewas no specific objection to the arguments or calculations in the submitted paper.The author pointed out to the editor the absence of any substantive objections bythe referee. The editor, Subrahmayan Chandrasekhar, sent the paper to a secondreferee. Sometime in the summer the second referee responded that the paper wasmisguided and recommended against publication, again with no specific criticismexcept that the author was obviously unfamiliar with the literature. Again, theauthor’s response to the editor was to note that there was no substantive criticism,no specific fault pointed out. Some days later Chandrasekhar appeared in theauthor’s office with the paper in his hand and said something along the lines of,‘Now see here, Parker, do you really want to publish this paper?’ I replied thatI did. Whereupon he said, ‘I have sent it to two eminent experts in the field andthey have both said that the paper is misguided and completely off the mark.’ Ireplied that my problem with the referees was that they were clearly displeased,but had nothing more to say. Chandrasekhar was silent for a moment and thanhe said, ‘All right, I will publish it.’ And that is how the paper without the words‘solar wind’ finally appeared in the November issue of the Astrophysical Journal.”Just about the same time when Parker (1958) published his paper solving theBiermann-Chapman puzzle, Alfv´en (1957) pointed out that the plasma densitiesinferred by Biermann (1951) were inconsistent with observed coronal densities (assuming that the plasma density decreases with the square of the heliocentricdistance as the solar corpuscular radiation moves outward). Alfv´en (1957) offeredan alternative explanation for the formation of cometary plasma tails that is de-picted in Fig. 4. The main assumption in this model was that the solar corpuscularradiation was carrying a “frozen-in” magnetic field that “hangs up” in the high
B a) b)c) d)
Fig. 4
Alfv´en’s scenario of the formation of comet tails. A plasma beam with a frozen-inmagnetic field approaches the head of a comet (panel a); the field is deformed (panels b andc); the final state is reached when the beam has passed (panel d) (Alfv´en 1957). density inner coma, where the solar particles strongly interact with the cometaryatmosphere and consequently the solar plasma flow considerably slows down. Thisinteraction results in a “folding” of the magnetic field around the cometary comathat creates the long plasma tail. Disturbances along the folded magnetic fieldlines propagate as magnetohydrodynamic waves and they can reach velocities of100 km/s even if the solar plasma has a density of ∼ − . It is interesting tonote that Alfv´en accepted Biermann’s (1951) idea of the continuous plasma out-flow from the Sun and naturally concluded that this plasma outflow must carry anembedded magnetic field, but he did not worry about the origin of such a plasmaflow.2.3 Solar breeze?Parker’s (1958) paper generated swift negative reaction. The community was notready to give up the idea of a hydrostatic corona and accept a continuously escap-ing solar atmosphere.The most prominent critic of Parker was Joseph Chamberlain, who received his PhD from the University of Michigan in 1952. At that time he was on thefaculty of the University of Chicago, the same institution where Chandrasekharand Parker worked. In September 1959, less than a year after Parker’s (1958) paperwas published, he submitted a paper which strongly criticized Parker’s (1958) solarwind solution. Chamberlain (1960) pointed out that Eq. (1) had two solutions that xtended MHD modeling of the steady solar corona and the solar wind 11 P a r k e r C h a m b e r l a i n Fig. 5
Comparison of the Parker (1958) and Chamberlain (1960) solar corona solutions. met the condition that the velocity was small at the coronal base. In addition toParker’s (1958) solution, a second solution described a solar corona that startedto expand, but the expansion gradually stopped beyond the critical point ( r c = av esc / v m ). A comparison of the Parker (1958) and Chamberlain (1960) solutionsis shown in Fig. 5.Chamberlain’s (1960) solution provided a hydrostatic coronal density distribu-tion and the expansion velocity returned to zero at infinity. This solution “saved”most features of the prevailing hydrostatic atmosphere model and it was greetedwith great relief by most of the solar community. The disagreement between Parkerand Chamberlain became quite heated. In 1959 when the debate took place Parkerwas a junior (untenured) faculty member at the University of Chicago and, as aresult of the ongoing controversy with a more senior faculty member, his tenurecase became more challenging. The controversy was eventually resolved – at leastin the eyes of the space physics community – by the beginning of the space age,when in-situ observations proved the existence of the solar wind.Gringauz et al (1960) analyzed the results of the Lunik-2 spacecraft and de-termined an interplanetary corpuscular particle flux of about 2 × ions/cm /s:“Starting from 9.30 hr Moscow time on 13 September 1959 up to the momentof the container of the second space rocket reaching the moon the container wasrecorded as passing through a positive ion flux (in all probability protons) withenergies exceeding 15 eV; Φ ∼ × ions/cm /s.” In a May 1962 presentation inWashington, Gringauz et al (1964) used observations of the first deep space probe(Venera-1) to estimate the speed of the solar corpuscular radiation to be about 400km/s. In addition, measurements by Bonetti et al (1963) on Explorer 10 confirmedthese initial results. However, because the Lunik, Venera and Explorer observa- tions were short-term, the doubters had some wiggle-room and the observationswere not regarded as definitive.It was not until the end of 1962, when the first Mariner 2 results were reported(Neugebauer and Snyder 1966), that the existence of the solar wind was widelyaccepted. The observed solar wind had typical proton densities of 5 to 20 cm − and Aug 27 Sep 1 Sep 6 Sep 11 Sep 16 Sep 21Sep 23 Sep 28 Oct 3 Oct 8 Oct 13 Oct 18Oct 20 Oct 25 Oct 30 Nov 4 Nov 9 Nov 14Nov 16 Nov 21 Nov 26 Dec 1 Dec 6 Dec 11Dec 13 Dec18 Dec 23 Dec28600 –500 –700 –400 –300 –600 –500 –700 –400 –300 –600 –500 –700 –400 –300 –600 –500 –700 –400 –300 –600 –500 –700 –400 –300 – V e l o c i t y , k m / s - 100- - 10- - 1- 100- - 10- - 1- 100- - 10- - 1- 100- - 10- - 1- 100- - 10- - 1 p r o t o n s / c m Rotation 1767Rotation 1768Rotation 1769Rotation 1770Rotation 1771
Mariner-2 Measurements in 1962 (Neugebauer & Snyder, JGR, 71, 4469, 1966)
Fig. 6
Mariner-2 observations of the continuous solar wind (Neugebauer and Snyder 1966). velocities between 300 and 700 km/s (see Fig. 6). The observed temperature was inthe range of 3 × –3 × K. These observations confirmed the predictions of theBiermann (1951)–Alfv´en (1957)–Parker (1958) theory and proved that the conceptof a static, slowly evaporating corona was incorrect (Chapman 1957; Chamberlain1960).It is interesting to note Chamberlain’s reaction to the eventual acceptance of the solar wind concept. Even some thirty years later, he tried to show that thosewho advocated the solar wind concept were right for the wrong reasons and theyadvocated inappropriate solutions. Here is a quote from Chamberlain’s 1995 paper:“In the early days of solar-wind theory, Parker (1958) appeared to be influencedby two seminal hypotheses: (a) Biermann’s conclusion that the behavior of comet xtended MHD modeling of the steady solar corona and the solar wind 13 tails was governed more by the interplanetary medium than by solar-radiationpressure, and (b) Chapman’s advocacy of a static solar corona that was heated togreat distances by conduction. Parker showed that a static corona was untenableand then constructed a primitive hydrodynamic model, which he labeled the solarwind , that would account for Biermann’s analysis of comet tails. I describe thissolar-wind model as ‘primitive’ because its temperature distribution, instead ofbeing derived physically, was characterized by a polytropic index, which simplifiedthe model enormously.At this point I was skeptical that Parker’s supersonic solutions were realistic (Chamberlain 1960) and, to investigate the problem, I wrote down the three equa-tions – now called the solar-wind equations – for a hydrodynamic solar corona thatwas heated from below by thermal conduction (Chamberlain 1961). I restrictedmy investigation to slow (subsonic) expansion, or models in which the parame-ter ε (described below) was 0. ln a jocular spirit, I referred to those solutionsas solar-breeze models , and I suggested that they were merely the hydrodynamiccounterpart to Waterston-Jeans evaporative escape.Shortly afterward, a solar-wind model, based on the same three equations butencompassing supersonic expansion, was developed by Scarf and Noble. (The pairof papers, Noble and Scarf (1963) and Scarf and Noble (1965), are hereinafterabbreviated by ‘SN-I’ and ‘SN-II,’ respectively.) But their model, although quiteacceptable as an illustrative case, is not physically accurate, even within the con-straints of its own simplifying assumptions.” v ( r ), reached the value of (cid:112) kT ( r ) /m p .In order to avoid numerical problems, Noble and Scarf (1963) integrated the equa-tions from the Earth inward. This was made possible by the fact that at that timethe solar wind conditions were observed. Noble and Scarf (1963) chose 1 AU valuesof n = 3 . − , v = 352 km/s, and T = 2 . × K.A typical solution without heat conduction and viscous effects is shown inFig. 7. We note that even in this relatively simple case the transonic solar windsolution describes the observed electron density profile within a factor of 2 or 3, asurprisingly good agreement. Scarf and Noble (1965) also considered the “subsonicsolution,” representing the solar breeze (Chamberlain 1960). When they took into consideration heat conduction and viscosity, Scarf and Noble (1965) were able toget excellent agreement with observations (however, in effect, they increased thenumber of free parameters, so the improved agreement is not really surprising). Asimilar solution with heat conduction but without viscosity was also obtained byWhang and Chang (1965).
Fig. 7
Comparison between simulated and observed electron densities in the inner corona(neglecting heat conduction and viscous effects) (Scarf and Noble 1965). T e and T i ) can deviate from each other. They devel-oped a two-temperature model where the plasma remains quasi-neutral ( n e ≈ n i )and current-free ( u e = u i ), but the two temperatures can be different. Sturrockand Hartle (1966) still considered a spherically symmetric problem and neglectedmagnetic field effects, but their model represented a step forward. By combin-ing the continuity, momentum, and energy equations they derived separate “heatequations” for electrons and ions (Sturrock and Hartle 1966):32 1 T s dT s dr − n dndr = 1 ΦkT s ddr (cid:18) r κ s dT s dr (cid:19) + 32 ν ei v T t − T s T s , (4) where s = e, i refers to either electrons or ions, the t subscript refers to the otherspecies, Φ = nvr is the spherical particle flux, v is the plasma flow speed, κ s isthe heat conductivity of the appropriate species and ν ei is the electron-ion energytransfer collision frequency. Sturrock and Hartle (1966) used the Chapman (1954)approximation for the collision frequency ( ν ei = 8 . × − nT − / e , where n is xtended MHD modeling of the steady solar corona and the solar wind 15 given in units of cm − ) and the Spitzer (1962) conductivities ( κ e = 6 × − T / e s − and κ i = 1 . × − T / i s − ). Fig. 8
Left panel: Flow velocity (solid line) and electron density (broken line) as a functionof radial distance (in units of solar radii) from the center of the Sun. Right panel: Electron(broken line) and ion (solid line) temperatures as a function of radial distance (Sturrock andHartle 1966).
Figure 8 shows the two-temperature solar wind solution (Sturrock and Hartle1966). At Earth orbit they obtained v = 270 km/s, n = 13 cm − , T i = 2800 K and T e = 4 . × K. While these values were in the right ballpark, the wind speedwas too slow, the density was too high, and the two temperatures were quite abit off ( T e too high and T i too low). The authors lamented that “We are left withthe problem of understanding why the solar-wind parameters do not always agreewith our model.” In other words, it was Nature’s fault that it did not agree withthe predictions of the model...3.3 Potential magnetic fieldBy the late 1960s, it has become clear that the magnetic field plays a major rolein determining the density as well as the velocity and temperature structure ofthe corona. The first models used the observed line-of-sight component of thephotospheric magnetic field to determine the magnetic field of the solar corona inthe current-free (or potential-field) approximation (Schatten 1968, 1969; Schattenet al 1969; Altschuler and Newkirk 1969; Newkirk and Altschuler 1970; Schatten1971).The potential field model is based on the fundamental assumption that themagnetic field above the photosphere is current free ( ∇ × B = 0 when r > R (cid:12) )and therefore the coronal magnetic field can be represented by a magnetic scalar potential, ψ : B = −∇ ψ . (5)Since there are no magnetic monopoles ( ∇ · B = 0) we obtain ∇ ψ = 0 . (6) The solution of Eq. (6) can be written as an infinite series of spherical har-monics: ψ ( r, θ, φ ) = R (cid:12) ∞ (cid:88) n =1 n (cid:88) m =0 (cid:18) R (cid:12) r (cid:19) n +1 [ g mn cos mφ + h mn sin mφ ] P mn ( θ ) , (7)where the coefficients g mn and h mn need to be determined from solar line-of-sightobservations and P mn ( θ ) are the associated Legendre polynomials with Schmidtnormalization:14 π (cid:90) π dθ (cid:90) π dφP mn ( θ ) P m (cid:48) n (cid:48) ( θ ) (cid:26) cos mφ sin mφ (cid:27) (cid:26) cos m (cid:48) φ sin m (cid:48) φ (cid:27) sin θ = 12 n + 1 δ nn (cid:48) δ mm (cid:48) . (8)With the help of the magnetic scalar potential, the magnetic field vector can beobtained anywhere above the photosphere: B = ( B r , B θ , B φ ) = (cid:18) − ∂ψ∂r , − r ∂ψ∂φ , − r sin θ ∂ψ∂φ (cid:19) . (9)It was recognized by Schatten et al (1969) that the coronal magnetic field fol-lows the current-free potential solution between the photosphere and a “sourcesurface” (located at r = R s ) where the potential vanishes and the magnetic fieldbecomes radial. This requires a network of thin current sheets for r ≥ R s (cf.Schatten 1971). Such a potential field can be described with the help of Legendrepolynomials that define the magnetic potential between two spherical shells, lo-cated at r = R (cid:12) and r = R s , each containing a distribution of magnetic sources(Altschuler and Newkirk 1969): ψ s ( r, θ, φ ) = R (cid:12) ∞ (cid:88) n =1 f n ( r ) n (cid:88) m =0 [ g mn cos mφ + h mn sin mφ ] P mn ( θ ) , (10)where f n ( r ) = (cid:16) R s R (cid:12) (cid:17) n +1 (cid:16) R s R (cid:12) (cid:17) n +1 − (cid:18) R (cid:12) r (cid:19) n +1 − (cid:16) R s R (cid:12) (cid:17) n +1 − (cid:18) rR (cid:12) (cid:19) n . (11)For the optimal radius of the source surface Schatten et al (1969) found R s =1 . R (cid:12) , while Altschuler and Newkirk (1969) recommended R s = 2 . R (cid:12) . Today,most potential field source surface (PFSS) models use the R s = 2 . R (cid:12) value. Wenote that at the source surface f n ( R s ) = 1 (cid:16) R s R (cid:12) (cid:17) n +1 − (cid:18) R s R (cid:12) (cid:19) n − (cid:16) R s R (cid:12) (cid:17) n +1 − (cid:18) R s R (cid:12) (cid:19) n = 0 . (12)With the help of expression (10), the magnetic field components can be written as B r ( r, θ, φ ) = (cid:18) R (cid:12) r (cid:19) ∞ (cid:88) n =1 n f n ( r ) n (cid:88) m =0 [ g mn cos mφ + h mn sin mφ ] P mn ( θ )+ ∞ (cid:88) n =1 (cid:16) R s R (cid:12) (cid:17) n +1 (cid:16) R s R (cid:12) (cid:17) n +1 − (cid:18) R (cid:12) r (cid:19) n +2 n (cid:88) m =0 [ g mn cos mφ + h mn sin mφ ] P mn ( θ ) (13) xtended MHD modeling of the steady solar corona and the solar wind 17 B θ ( r, θ, φ ) = − (cid:18) R (cid:12) r (cid:19) ∞ (cid:88) n =1 f n ( r ) n (cid:88) m =0 [ g mn cos mφ + h mn sin mφ ] dP mn ( θ ) dθ (14) B φ ( r, θ, φ ) = (cid:18) R (cid:12) r sin θ (cid:19) ∞ (cid:88) n =1 f n ( r ) n (cid:88) m =0 m [ g mn sin mφ − h mn cos mφ ] P mn ( θ ) (15)At the source surface the magnetic field becomes radial because f n ( R s ) = 0 and B r ( R s , θ, φ ) = ∞ (cid:88) n =1 (cid:16) R s R (cid:12) (cid:17) n − (cid:16) R s R (cid:12) (cid:17) n +1 − n (cid:88) m =0 [ g mn cos mφ + h mn sin mφ ] P mn ( θ ) . (16) Fig. 9
Magnetic field line map obtained with the potential field source surface (PFSS) model(Altschuler and Newkirk 1969).
Outside the source surface it is assumed that the radial flow of the solar windcarries the magnetic field outward into the heliosphere. This region is not describedby the PFSS model. Between the photosphere and the source surface the magneticfield vector components can be described by expressions (13), (14) and (15). Theinner boundary condition at the photosphere is obtained from the observed line-of-sight magnetic field components using a least square fit (cf. Hoeksema et al 1982).
These measurements are used to determine the expansion coefficients g mn and h mn .The outer boundary condition at the source surface is that the field is normal tothe source surface, consistent with the assumption that it is then carried outwardby the solar wind. This condition is automatically satisfied by our selection of the f n function (Eq. (11)). An example of the PFSS solution is shown in Fig. 9. Fig. 10
Analytic solution of a helmet configuration (Pneuman 1968).
Nearly simultaneously with the development of the source surface models theeffect of the coronal expansion on the magnetic field was also explored (Pneuman1966, 1968, 1969; Pneuman and Kopp 1971). In a series of papers Pneuman (1966,1968, 1969) analytically investigated how centrifugal, pressure gradient, and mag-netic forces impact the flow of an infinitely conducting fluid where the magneticfield lines and plasma flow lines must coincide in the corotating frame.An early example of a numerical model of the corona with solar wind is byPneuman and Kopp (1971). As in the Parker solution (Parker 1963), this modelpossesses the high temperature (
T > ρ ≈ − gcm − ) plasma whose pressure cannot be held in equilibrium bysolar gravity or the pressure of the interstellar medium. Consequently, the coronalplasma expands rapidly outward achieving supersonic speeds within a few solarradii, and in doing so forms the solar wind.Early numerical models prescribed volumetric coronal heating in ways thatstrived to accounted for the effects thermal conduction and radiative losses, aswell as satisfying known constraints of coronal heating. However, these works foundthat the observed fast solar wind (speed 700-800 km/s) cannot be produced bythermal pressure without temperatures greatly exceeding coronal values, particu-larly in coronal holes where the fast wind originates.Using simplified conservation laws, Pneuman (1966, 1968, 1969) obtained helmet- like closed field line regions buffeted by converging open field lines (Fig. 10). Later,Pneuman and Kopp (1971) numerically solved the conservation equations for anaxially symmetric steadily expanding corona with an embedded magnetic dipole,assuming North-South symmetry. In this case the solution is not only axially sym-metric, but it is also symmetric with respect to the equatorial plane. xtended MHD modeling of the steady solar corona and the solar wind 19 Fig. 11
A comparison of the magnetic field (solid curves) with a potential field (dashedcurves) having the same normal component at the reference level. The field lines for the twoconfigurations are chosen so as to be coincident at the surface. (Pneuman and Kopp 1971).
The Pneuman and Kopp (1971) solution shows several interesting features. Animportant aspect of the solution is the development of a neutral point at the topof the helmet where the magnetic field vanishes. A current sheet forms betweenthe regions of opposite magnetic polarity above the neutral point. The outflowingplasma reaches the Alfv´en velocity just above the neutral point (at the top of thehelmet), so the top of the helmet is, in effect, the transition from sub-Alfv´enic tosuper-Alfv´enic flow.Fig. 11 shows a comparison of the field lines of the numerical solution anda potential field model with the same normal component at the reference level(Pneuman and Kopp 1971). To make the comparison meaningful, the field linesfor the two cases are chosen so as to be coincident at the surface. As expected,the field is everywhere stretched outward by the gas with this distention becominglarge near the neutral point. In the closed region well below the cusp the differ-ence between the two configurations is small, mainly because the pressure at thereference level is taken to be independent of latitude. The outward distention ofthe field in this region, as a result, is produced solely by currents generated by theexpansion along open field lines. For the general case of variable surface pressure,however, a pressure and gravitational force balance cannot be satisfied normal to the field and significant j × B forces are expected to maintain the equilibrium.In spite of its obvious limitations, the Pneuman and Kopp (1971) simulationestablished the usefulness of MHD simulations of the solar corona. It was the firstsuccessful attempt to apply the conservation laws of magnetohydrodynamics toexplain large-scale features of the solar corona. We adopt the “ambient-transient” paradigm to modeling the dynamic and highlystructured solar wind. By “ambient” we mean a quiet-sun-driven full 3-D struc-ture for the interplanetary magnetic field and a 3-D distribution of the solar windparameters. They are both close to being steady-state in the frame of reference co-rotating with the Sun, except for the highly intermittent solar wind region. Thisambient solution determines the bimodal structure of the solar wind. It affectsmagnetic connectivity between the active regions at the Sun and the correspond-ing regions in the heliosphere. In turn such connectivity affects energetic particleacceleration and transport.As we discussed in Section 3, the first generation of magnetohydrodynamicmodels of the interplanetary medium were developed in the second half of the 1970sand were used for about two decades (Steinolfson et al 1975, 1978; Pizzo 1978, 1980,1982; Steinolfson et al 1982; Steinolfson 1988; Steinolfson and Hundhausen 1988;Pizzo 1989; Steinolfson 1990; Pizzo 1991; Steinolfson 1992; Pizzo et al 1993; Pizzoand Gosling 1994; Pizzo 1994b,a; Steinolfson 1994). These models were designedto describe only large-scale bulk-average features of the plasma observed throughthe solar cycle. At solar minimum, these coronal structures are the following:1. open magnetic field lines forming coronal holes;2. closed magnetic field lines forming a streamer belt at low latitudes;3. the bimodal nature of the solar wind is reproduced with fast wind originatingfrom coronal holes over the poles and slow wind at low latitudes.A thin current sheet forms at the tip of the streamer belt and separates oppositedirected magnetic flux originating from the two poles. At solar minimum, thefast wind lies at 30 degrees heliographic latitude and has an average velocity of750 km s − at distances greater than 15 solar radii, at which distance the windhas attained the majority of its terminal velocity. The slow wind, by contrast, isconfined close to the global heliospheric current sheet, which lies near the equatorat solar minimum. This component of the wind is highly variable, with speedsthat lie between 300 and 450 km s − . The slow solar wind has been suggestedby Wang and Sheeley (1990) to originate from highly expanding plasma travelingdown magnetic flux tubes that originate near coronal hole boundaries. It has alsobeen suggested that opening of closed flux tubes by interchange reconnection withopen flux may release plasma to form the slow solar wind (Fisk et al 1998; Lionelloet al 2005; Rappazzo et al 2012). More recent theories have related the slow solarwind to complex magnetic topology flux tubes near the heliospheric current sheet,which are characterized by the squashing factor (Titov et al 2012; Antiochos et al2012). At solar maximum, the current sheet is highly inclined with smaller coronalholes forming at all latitudes, while the fast wind is largely absent. ρ , v r , v φ , p , B r , and B φ as a function of time andradial distance. The model was applied to a forward-reverse shock pair propagating xtended MHD modeling of the steady solar corona and the solar wind 21 in this simplified solar wind and the solution was compared with the results ofsimilarity theory. In the end Steinolfson et al (1975) concluded that MHD effectsmight be important in the dynamical behavior of the solar wind near Earth orbit.
1 2 3 4 5 B fi e l d l i n e s v e l o c i t y v e c t o r s A l f v é n s u r f a c e s o n i c s u r f a c e Fig. 12
Steady-state coronal structure for a plasma β value of 0.5. Solid lines representmagnetic field lines, while arrows show plasma flow velocity vectors. The horizontal axis is thesolar equator (Steinolfson et al 1982). In a follow-up paper, Steinolfson et al (1978) considered a different situation,when the flow and magnetic field are in the meridional plane. They solved timeevolution equations for ρ , v r , v θ , p , B r , and B θ as a function of polar angle andradial distance. They considered solar transients in two magnetic configurations:with the magnetic field radial (open), and with the magnetic field parallel to thesolar surface (closed). The solar event is simulated by a pressure pulse at the base ofan initially hydrostatic atmosphere. The pressure pulse ejects material into the lowcorona and produces a disturbance that propagates radially and laterally throughthe corona. The disturbance is preceded by waves (which may strengthen intoshocks) that travel in the meridional plane with the shape of an expanding loop.The portion of the disturbance between the ejected material and the precedingwaves consists entirely of coronal material whose properties have been altered bythe waves. This simulation, in spite of its many limitations, was the first successful attempt to simulate coronal transients.A few years later, Steinolfson et al (1982) revisited the steady global solarcorona that was investigated a decade earlier by Pneuman and Kopp (1971). Thismore sophisticated simulation allowed for non-constant temperature and did notrequire specific conditions to be met at the cusp (Pneuman and Kopp 1971). They applied their previous model (Steinolfson et al 1978) to study the axiallysymmetric global corona. Their solution essentially confirmed the configurationobtained by Pneuman and Kopp (1971) with some notable differences. Since thecoronal plasma is subsonic and sub-Alfv´enic near the Sun the flow is considerablymore complex than predicted by Pneuman and Kopp (1971). The main feature,however, is essentially the same: the coronal outflow results in closed field linesnear the equator and open field lines (coronal holes) at high latitudes (see Fig. 12).In parallel with Steinolfson’s efforts, Pizzo took a very different approach tosimulating the interplanetary medium (Pizzo 1978, 1980, 1982, 1989, 1991; Pizzoet al 1993; Pizzo and Gosling 1994; Pizzo 1994a,b). As a first step, Pizzo (1978)developed a 3D hydrodynamic model of steady corotating streams in the solarwind, assuming a supersonic, inviscid and polytropic flow beyond approximately35 R s . This approach takes advantage of the fact that in the inertial frame thetemporal and azimuthal gradients are related by ∂ t = − Ω (cid:12) ∂ φ , where φ is theazimuth angle and Ω (cid:12) is the equatorial angular velocity of the Sun.The steady-state conservation equations were solved using an explicit Eulerianapproach on a rectangular ( θ, φ ) grid covering the 45 ◦ ≤ θ ≤ ◦ and − ◦ ≤ φ ≤ ◦ spherical surface. Periodic boundary conditions were imposed at the azimuthaledges of the mesh, while the latitudinal boundaries are free surfaces, with themeridional derivatives approximated by one-sided differences. The density, velocityvector, and scalar pressure were defined at the inner boundary (35 R (cid:12) ). Knowingthe solution at a heliocentric distance r , the solution was advanced to r + ∆r usingthe conservation equations. Using a value of ∆r = 30km, Pizzo (1978) obtained asteadily corotating stream structure between 35 R (cid:12) and 1AU.Specifically, Pizzo (1978, 1980) solved the governing equations that describethe dynamical evolution of 3-D corotating solar wind structures. The model islimited to those structures that are steady or nearly steady in the frame rotatingwith the Sun and utilizes the single-fluid, polytropic, nonlinear, 3D hydrodynamicequations to approximate the dynamics that occur in interplanetary space, wherethe flow is supersonic and the governing equations are hyperbolic. In the inertialframe, the equations are the following (Pizzo 1978, 1980): − Ω (cid:12) ∂ρ∂φ + ∇ · ( ρ u ) = 0 (17) − Ω (cid:12) (cid:18) e r ∂u r ∂φ + e θ ∂u θ ∂φ + e φ ∂u φ ∂φ (cid:19) + ( u · ∇ ) u = − ρ ∇ p − GM (cid:12) r e r (18) (cid:18) − Ω (cid:12) ∂∂φ + u · ∇ (cid:19) (cid:18) pρ γ (cid:19) = 0 , (19)where ρ is the mass density, u is the center of mass velocity, p is the total isotropic(scalar) gas pressure, G is the gravitational constant, M (cid:12) is the solar mass, γ is thepolytropic index and Ω (cid:12) is the equatorial angular rotation rate of the Sun. The independent variables are the spherical polar coordinates ( r, θ, φ ). Conduction,wave dissipation, differential rotation, the magnetic field, and shock heating areall neglected. Equations (17–19) can be rearranged to obtain a set of differentialequations describing the radial evolution of the primitive variables and solved bymarching in radial distance from the inner boundary outward. This method yields xtended MHD modeling of the steady solar corona and the solar wind 23 Fig. 13
Equatorial solution for an initially steep-sided circular stream at I AU. F and R markthe forward and reverse wave fronts, which demarcate the dynamic compression ridge. Thetwo shocks propagate in opposite directions, but both are convected outward in the bulk flow.The interface, I, is a shear layer separating fast and slow flow regimes that initially had verydifferent fluid properties (Pizzo 1982). a 3D solution that is steady-state in the corotating frame and describes the streamstructure in the interplanetary medium.In a subsequent paper, Pizzo (1982) extended his marching method to includethe interplanetary magnetic field. He considered the implication of a high-speedstream emanating from the Sun and expanding to 1 AU. The central portion ofthe stream is a circular plateau some 30 ◦ in diameter where the radial velocity isa uniform 600 km/s. The speed falls off smoothly in all directions, bottoming outat the background speed of 300 km/s at a distance of 7.5 ◦ from the periphery of the plateau. The results at 1 AU are shown in Fig. 13. The panels from top tobottom show the bulk speed, the flow angle in the inertial frame, the flow angle inthe rotating frame, the number density, the single-fluid temperature, the magneticfield intensity and the total pressure as a function of azimuth. At the front ofthe stream (west, or left), forward (F) and reverse (R) waves have formed about an interface (I). The two waves, which propagate in opposite directions from theinterface, originate in the general compression at the stream front near the Sun.The Pizzo (1982) model was later applied to qualitatively explain Ulysses ob-servations at larger helio-latitudes (Pizzo and Gosling 1994). Ulysses discoveredthat the forward-reverse shock pair structure commonly bordering corotating in-teraction regions beyond 1 AU near the ecliptic plane undergoes a profound changenear the maximum heliographic latitude of the heliospheric current sheet. At thislatitude the forward shock is observed to weaken abruptly, appearing as a broadforward wave, while the reverse shock weakens much more slowly (Gosling et al1993). These observations were well reproduced by the Pizzo (1982) model (Pizzoand Gosling 1994).4.2 Connecting the corona and the heliosphereProbably the most challenging region to model is the transition from the cold( ∼ ∼
500 km ( < − R (cid:12) ). The firstattempts to include this physics in 3D simulations are just beginning (see Section5 in this paper and Sokolov et al 2016).In the early 1990s it was recognized that the solar wind speed at 1 AU nega-tively correlates with a magnetic flux tube expansion factor near the Sun (Wangand Sheeley 1990, 1992, 1995). This expansion factor describes the ratio of a givenflux tube’s cross sectional area at some heliocentric distance and the cross sectionalarea of the same flux tube at the solar surface. Wang and Sheeley (1990, 1992,1995) found that the solar wind speed at a heliocentric distance can be expressedas u ( r ) = u min + u max − u min f exp ( r ) α , (20)where u min and u max represent the slow and fast solar wind speeds, f exp is theexpansion factor at heliocentric location r , while α is an empirical factor (nearunity). Later Arge and Pizzo (2000) and Arge et al (2004) generalized the Wangand Sheeley (1990, 1992, 1995) formula and introduced several additional param-eters. A comprehensive description of the Wang-Sheeley-Arge (WSA) model andits parameter values can be found in Riley et al (2015).The WSA formula provides an efficient and simple way to circumvent thecomplex physics of the low corona and transition region. It is used by a numberof heliosphere models to provide inner boundary conditions beyond the Alfv´ensurface (where the solar wind speed exceeds the local Alfv´en speed). These modelstypically place their inner boundaries in the 20 to 30 R (cid:12) range (e.g., Wold et al2018). xtended MHD modeling of the steady solar corona and the solar wind 25 (CME) propagation (Odstrˇcil and Pizzo 1999; Odstrcil et al 2004, 2005; Siscoeand Odstrcil 2008). ENLIL has been adapted to accept inner boundary solar windconditions from a variety of sources, including the WSA model (Wang and Shee-ley 1990; Arge et al 2004) and coronal MHD models (Odstrˇcil et al 2002, 2004).In the case of Hayashi (2012), the inner boundary conditions (outside the criti-cal point) are derived from interplanetary scintillation (IPS) observations (Jack-son et al 1998). More recently, other groups also developed 3D inner heliospheremodels with super-Alfv´enic inner boundary conditions using the WSA approach.These include the LFM-Helio model (Merkin et al 2011, 2016), the SUSANOO-SWcode (Shiota et al 2014; Shiota and Kataoka 2016), the MS-FLUKSS suite (Kimet al 2016) and EUHFORIA developed by the Leuwen group (Pomoell and Poedts2018).WSA-like empirical inner boundary conditions were also used in the first gen-eration of outer heliosphere models describing the interaction between the solarwind and the interstellar medium. Linde et al (1998) published the first 3D MHDmodel describing the interaction of the magnetized solar wind with the magne-tized interstellar medium. This simulation also took into account the presence ofthe neutral component of the interstellar medium and the resulting mass loadingprocess inside the heliosphere. Fig. 14 presents a 3D view of the global heliosphere. VB Fig. 14
Three-dimensional view of the global heliosphere. The color code shows the log ofplasma density. Yellow lines are the plasma velocity streamlines and the white lines follow themagnetic field lines. Black arrows indicate the direction of the magnetic field along a cross-tailcut placed 225 AU downstream from the Sun. (from Linde et al 1998)
Following the early work of Linde et al (1998), several groups developed sophis-ticated 3D models of the outer heliosphere and its interaction with the magnetizedinterstellar medium. While details of these models go beyond the scope of this re-view we note the progress made by Opher et al (2003, 2006, 2007, 2016) and theHuntsville group (Pogorelov et al 2013, 2015; Zirnstein et al 2017; Pogorelov et al2017). / ad-hoc heating rate that is chosen to fit observations; or (ii) include a semi-empirical coronal heating function that is based on the physics of Alfv´en waves.Examples for the first approach include papers by Groth et al (1999a, 2000a);Lionello et al (2001b, 2009a); Riley et al (2006); Feng et al (2007); Nakamizo et al(2009); Feng et al (2010); Titov et al (2008), and Downs et al (2010). An importantlimitation of this approach is that models utilizing an ad-hoc approach depend onsome free parameters that need to be determined for various solar conditions.While the ad-hoc heating function approach is well-suited for typical conditions, itcan’t properly account for unusual conditions, such as those that can occur duringextreme solar events.A major benefit of the WSA model is its simplicity and therefore it can beseamlessly integrated into global-scale space weather simulations (e.g., Odstrˇcil2003; Cohen et al 2007). In the Cohen et al (2007) study the WSA formulaewere used as the boundary condition for a large-scale 3-D MHD simulation withvaried polytropic gas index distribution (see Roussev et al 2003). These modelscan successfully reproduce observed solar wind parameters at 1 AU.An example of a solar model driven by empirical heating is shown in Fig. 15,which depicts solar minimum conditions on a meridional plane close to the Sun.This result from Manchester et al (2004) is based on a model by Groth et al(2000a). Here the color image indicates the velocity magnitude, | u | , of the plasmawhile the magnetic field is represented by solid white lines. Inspection reveals abimodal outflow pattern with slow wind leaving the Sun below 400 km/s nearthe equator and high-speed wind above 700 km/s found above 30 ◦ latitude. Inthis model, we find that the source of the slow solar wind is plasma originatingfrom the coronal hole boundaries that over-expands and fails to accelerate to highspeed as it fills the volume of space radially above the streamer belt. This modelis consistent with the empirical model of the solar wind proposed by Wang andSheeley (1994) that explains solar wind speeds as being inversely related to theexpansion of contained magnetic flux tubes. The magnetic field remains closedat low latitude close to the Sun, forming a streamer belt. At high latitude, themagnetic field is carried out with the solar wind to achieve an open configuration.Closer to the equator, closed loops are drawn out and, at a distance ( r > R (cid:12) ),collapse into a field reversal layer. The resulting field configuration has a neutral line and a current sheet originating at the tip of the streamer belt.Several validation and comparison studies have been published using this ap-proach (Owens et al 2008; V´asquez et al 2008; MacNeice 2009; Norquist and Meeks2010; Gressl et al 2014; Jian et al 2015; Reiss et al 2016). However, these modelsdo not capture the physics of Alfv´en wave turbulence or even neglect it altogether. xtended MHD modeling of the steady solar corona and the solar wind 27 Y [Rs] Z [ R s ] -10 -5 0 5 10-10-50510 |U| km/s700.0560.0420.0280.0140.00.0 Fig. 15
Magnetic structure and velocity for the steady-state solar wind solution by Manch-ester et al (2004). Solid white lines are magnetic “streamlines” drawn in the y − z planesuperimposed upon a false color image of the velocity magnitude. Note the bimodal nature ofthe solar wind. Even though some of these models were designed to account for the Alfv´en wavephysics (e.g., Cohen et al 2007), they do not capture many aspects of the interac-tion of the turbulence with the background plasma flow, which include both energyand momentum transfer from the turbulence to the solar wind plasma. BecauseAlfv´enic turbulence effects are likely to be of great significance in the near-Sundomain, these simpler models should be used with caution for simulating the solaratmosphere.4.5 Thermodynamic corona modelsWhile the polytropic solar corona models were quite successful in describing thequiet low latitude corona they had major problems in accounting for the two-state(slow and fast) solar wind and large dynamical processes (such as CMEs) that caninterrupt the quasi-steady state (in the corotating frame) situation. In particular,the shock thermodynamics got seriously distorted by the reduced adiabatic index and various spurious phenomena (like runaway plasma temperatures) appeared inthe solutions.In order to overcome this problem the full energy equation – with all accompa-nying computational challenges – needs to be considered. Miki´c et al (1999) pro-posed that the energy equation be solved with a realistic adiabatic index ( γ = 5 / Fig. 16
Magnetic field model for the Sun around the time of Whole Sun Month (CarringtonRotation 1913, 1996 August 22 to September 18). The simulation was carried out with thethermodynamic model described in Section 4.5. The magnetic flux distribution was projectedon the solar surface with selected magnetic field lines from the MHD solution. (from Lionelloet al 2009a) conduction, radiative cooling, and various heating processes. In particular, Miki´cet al (1999) introduced the following form of the heating function: S = −∇ · q − n e n p Q ( T ) + H ch + H d + D , (21)where H ch is the coronal heating source, D is the Alfv´en wave dissipation term, H d = ηJ + ν ∇ v : ∇ v represents heating due to viscous and resistive dissipation,and Q ( T ) is the radiative loss function. In the collisional regime (below ∼ R (cid:12) ),the heat flux is q = b ( b · ∇ ) T , where b is the unit vector along the magneticfield vector, and κ (cid:107) = 9 × T / is the Spitzer value of the parallel thermal conductivity. The polytropic index γ is 5 /
3. In the collisionless regime (beyond ∼ R (cid:12) ), the heat flux is modeled by q = αn e kT v , where α is an empiricalparameter. The coronal heating source is a parameterized function given in theform of H ch ( r, θ ) = H ( θ ) exp (cid:20) − r − R (cid:12) λ ( θ ) (cid:21) , (22) xtended MHD modeling of the steady solar corona and the solar wind 29 where the empirical functions H ( θ ) and λ ( θ ) express the latitudinal variation ofthe volumetric heating and scale length, respectively.While this “thermodynamic” approach sidesteps the underlying physics of coro-nal heating and solar wind acceleration it provides an adequate mathematicalframework to describe the coronal processes in a way that is consistent with solardynamical processes.The thermodynamic coronal model has been successfully applied to simulatethe solar corona for the first whole Sun month (Lionello et al 2009a). The globalmagnetic configuration obtained with the model is shown in Fig. 16.4.6 Model inputsThe magnetic field is an essential component of the corona. As a matter of fact,the surface magnetic field distribution is the primary direct quantitative observ-able for models, physical variables associated with other observations need to beinferred. Early models used simple dipolar magnetic fields to simulate solar min-imum conditions (Groth et al 2000b; Steinolfson et al 1982). In this case, thedipole is chosen such that the maximum field strength at the poles approximately10 gauss. However, simulating realistic solar conditions requires the use of theobserved line-of-sight global magnetic field. To provide these data, full disk mag-netograms are taken as the Sun completes a full rotation, and then combined intoa full-surface synoptic map. Early examples of these maps include those providedby the Stanford Wilcox and Mount Wilson observatories as well as the GlobalOscillation Network Group (GONG). These data sources are further augmentedby offerings from the National Solar Observatory’s Synoptic Long-term Investiga-tion of the Sun (SOLIS), and NASA’s Solar Dynamics Observatory/Helioseismicand Magnetic Imager (HMI). Figure Fig. 17 provides an example input of a full-surface synoptic map based on GONG data. The use of synoptic maps in globalMHD models was pioneered by Wu et al (1999); Miki´c et al (1999); Linker et al(1999) and applied by many others (e.g., Roussev et al 2003; Hayashi 2005; Fenget al 2007; Cohen et al 2007; Nakamizo et al 2009; van der Holst et al 2010).While the line-of-sight photospheric magnetic field can be accurately measuredand extrapolated to coronal heights with reasonable accuracy, the base densitiesand temperatures remain much more difficult to accertain, especially on the globalscale required to specify boundary conditions for numerical models. In the case ofvan der Holst et al (2010), the base temperature and mass density were derivedempirically from the differential emission measure tomography (DEMT) techniqueby V´asquez et al (2008). While promising, this approach requires an involved, time-consuming calculation not suitable for operational use. In recent years, a physics-based approach has been used to self-consistently calculate the thermodynamicproperties at the base of the corona by applying field-aligned electron heat con- duction and radiative processes (typically derived from Chianti (Dere et al 1997)).With this approach, it is possible to self-consistently reproduce the transition re-gion. With that region, and the appropriate coronal density and temperatures,realistic heating functions can be applied (Lionello et al 2011; Sokolov et al 2013;van der Holst et al 2014). Carrington Longitude [Deg] L a t i t ud e [ D e g ] Br_0 [G]543210 1 2 3 4 5
Fig. 17
Carrington map of the radial magnetic field component at 1 R (cid:12) . This map is basedon a synoptic magnetogram of Carrington rotation 2109 (2011 April 12 to May 9) from GONGand processed to a PFSS solution using spherical harmonics. For the purpose of showing boththe coronal holes and active regions, the magnetic field in this plot is saturated by ±
3D coronal models applying synoptic magnetograms, in conjunction with ther- modynamic processes such as field-aligned heat conduction, are capable of bothpredicting and interpreting the detailed magnetic and plasma structure of thecorona. Successful comparisons began with the predicted appearance of the coronain Thomson-scattered white light, compared to coronagraph and eclipse images(Miki´c et al 1999). Figure Fig. 18 provides a very impressive prediction of white-light images of a solar eclipses (Miki´c et al 2007). Models capture the low-densitycoronal holes and high-density helmet streamers, including plasma sheets extend-ing into the low corona. Where models can reproduce coronal mass density andelectron temperature, they can predict thermal emission in the extreme ultravio-let and X-ray spectrum, and line-of-sight integration yields synthetic images thatshow reasonably good agreement with observations, as seen in Figure 19.When coronal models extend into the heliosphere, in particular beyond 1 AU,they offer predictions of the solar wind plasma parameters, including charge statecomposition that can be compared directly to in-situ observations. Early exam-ples include those by Wu et al (1999). Steady-state models can be validated byreplicating solar wind observations for the synodic rotation period measured atthe Earth, namely 27.27 days. Examples are shown in Cohen et al (2008); Fenget al (2012b); Meng et al (2015); T¨or¨ok et al (2018).4.8 Numerical mesh techniques
The computational domain for coronal simulations typically extends to r > R (cid:12) .Flows will be superfast at the outer boundary so that simple outflow boundaryconditions will be well prescribed. Most simulations use spherical grids with fixedangular resolution, but highly stretched in the radial direction to provide highresolution to the lower corona. In the case of Groth et al (2000b) and Manchester xtended MHD modeling of the steady solar corona and the solar wind 31 Fig. 18
Comparison between the MHD prediction (with magnetic field lines and polariza-tion brightness shown in the first two columns) and eclipse observations (shown in the thirdcolumn). (Miki´c et al 2007). et al (2004), an adaptive Cartesian grid was used with 4 × × / R (cid:12) to 2 R (cid:12) . Grids are spatially positioned tohighly resolve the corona as well as the heliospheric current sheet. More recentmodels have used spheric adaptive grids (e.g. Sokolov et al 2013; van der Holstet al 2014), while others have employed a cube-sphere to maintain the advantage of nearly constant angular resolution while avoiding the singularity at the poles.For example, Feng et al (2007, 2010, 2012a) developed the Solar-InterPlanetaryConservation Element/Solution Element (SIP-CESE) MHD model that employs asix-component Yin-Yang grid system similar to a cubed sphere grid, with sphericalshell-shaped domains extending from the low corona to 1 AU. Fig. 19
Top panels (left to right): observational images from SDO/AIA 211˚A, STEREOA/EUVI 171˚A, and STEREO B/EUVI 195˚A. The observation time is 2011 March 7 20:00UT (CR2107). Bottom panels: synthesized EUV images of the model. Active regions andcoronal holes are marked in both the observational and synthesized images, to demonstratethe reproducibility of the observed morphological structures in our simulations. (From Jin et al(2017)) in-situ space exploration. Examplesinclude papers by Belcher et al (1969), Belcher and Davis, Jr. (1971), and byAlazraki and Couturier (1971). A consistent and comprehensive theoretical de- scription of Alfv´en wave turbulence and its effect on the averaged plasma motionhas been developed in a series of works, particularly by Dewar (1970) and byJacques (1977, 1978) (see also references therein). More recent efforts to simulatesolar wind acceleration utilize the approach developed by Usmanov et al (2000).Currently, it is commonly accepted, that the gradient of the Alfv´en wave pressure xtended MHD modeling of the steady solar corona and the solar wind 33 is the key driver for solar wind acceleration, at least in fast flows. It is impor-tant to emphasize, that while incorporating the Alfv´en wave-driven accelerationis usually accomplished by including the wave pressure gradient in the governingequations (Jacques 1977), there is still no generally accepted approach to describethe coronal heating via the Alfv´en wave turbulence cascade.Fig. 20 shows the results of the first axisymmetric (2-D) simulation of thesolar corona and solar wind using self-consistent Alfv´en turbulence (Usmanov et al2000). After 64 hours of relaxation time, a closed field region develops near theequator; the flow velocity is high in the open polar field region but decreases towardthe equator above the region of the closed magnetic field.
Fig. 20
Magnetic field configuration near the Sun superimposed on a map of radial flowvelocities after 64 hours of relaxation. The Alfv´en line (where the radial flow velocity is equalto the Alfv´en velocity computed for the total magnetic field) is shown by the yellow line, andthe sonic line (where the radial velocity is equal to the sound velocity) is shown by the greenline. (Usmanov et al 2000).
Damping of Alfv´en wave turbulence as a source of coronal heating has alsobeen extensively studied from the early days of in situ solar wind observations(e.g., Barnes 1966, 1968). Later, it was demonstrated that reflection from sharppressure gradients in the solar wind (Heinemann and Olbert 1980; Leroy 1980) isa critical component of Alfv´en wave turbulence damping (Matthaeus et al 1999;Dmitruk et al 2002; Verdini and Velli 2007). For this reason, many numerical mod-els explore the generation of reflected counter-propagating waves as the underlying cause of the turbulence energy cascade (e.g., Cranmer and Van Ballegooijen 2010),which transports the energy of turbulence from the large-scale motions across the inertial range of the turbulence spatial scale to short-wavelength perturbations.The latter can be efficiently damped due to wave-particle interaction. In this way,the turbulence energy is converted to random (thermal) energy.
Many recent efforts aim to develop models that include Alfv´en waves as a pri-mary driving agent for both heating and accelerating of the solar wind. Examplesare papers by Hu et al (2003), Suzuki and Inutsuka (2005), Verdini et al (2010),Matsumoto and Suzuki (2012), and Lionello et al (2014a,b).5.2 Alfv´en wave turbulence driven solar atmosphere modelThe ad hoc elements can be eliminated from the solar corona model by assumingthat the coronal plasma is heated by the dissipation of Alfv´en wave turbulence(Sokolov et al 2013). The dissipation itself is caused by the nonlinear interactionbetween oppositely propagating waves (e.g., Hollweg 1986).Within coronal holes, there are no closed magnetic field lines, hence, thereare no oppositely propagating waves. Instead, a weak reflection of the outwardpropagating waves locally generates sunward propagating waves as quantified byvan der Holst et al (2014). The small power in these locally generated (and almostimmediately dissipated) inward propagating waves leads to a reduced turbulencedissipation rate in coronal holes, naturally resulting in the bimodal solar windstructure. Another consequence is that coronal holes look like cold black spots inthe EUV and X-ray images, while closed field regions are hot and bright. Activeregions, where the wave reflection is particularly strong, are the brightest in thismodel (see Sokolov et al 2013; Oran et al 2013; van der Holst et al 2014).The continuity, induction, and momentum equations used in the model are thefollowing: ∂ρ∂t + ∇ · ( ρ u ) = 0 , (23) ∂ B ∂t + ∇ · ( uB − Bu ) = 0 , (24) ∂ ( ρ u ) ∂t + ∇ · (cid:18) ρ uu − BB µ (cid:19) + ∇ (cid:18) p i + p e + B µ + p A (cid:19) = − GM (cid:12) ρ r r , (25)where ρ is the mass density, u is the bulk velocity ( u = | u | is assumed to be thesame for the ions and electrons), B is the magnetic field, G is the gravitationalconstant, M (cid:12) is the solar mass, r is the position vector relative to the centerof the Sun, µ is the magnetic permeability of vacuum. As has been shown byJacques (1977), the Alfv´en waves exert an isotropic pressure ( p A in the momentumequation). The relation between the wave pressure and wave energy density is p A = ( w + + w − ) /
2. Here, w ± are the energy densities for the turbulent wavespropagating along the magnetic field vector ( w + ) or in the opposite direction( w − ). The isotropic ion and electron pressures, p i and p e , are governed by theappropriate energy equations: ∂∂t (cid:18) p i γ − ρu B µ (cid:19) + ∇ · (cid:26)(cid:18) ρu γp i γ − B µ (cid:19) u − B ( u · B ) µ (cid:27) == − ( u · ∇ ) ( p e + p A ) + N e N i k B γ − (cid:18) ν ei N i (cid:19) ( T e − T i ) − GM (cid:12) ρ r · u r + Q i , (26) xtended MHD modeling of the steady solar corona and the solar wind 35 ∂∂t (cid:18) p e γ − (cid:19) + ∇ · (cid:18) p e γ − u (cid:19) + p e ∇ · u == −∇ · q e + N e N i k B γ − (cid:18) ν ei N i (cid:19) ( T i − T e ) − Q rad + Q e , (27)where T e,i are the electron and ion temperatures, N e,i are the electron and ionnumber densities, and k B is the Boltzmann constant. Other newly introducedterms are explained below.The ideal equation of state, p e,i = N e,i k B T e,i , is used for both species. Thepolytropic index is γ = 5 /
3. The optically thin radiative energy loss rate in thelower corona is given by Q rad = N e N i Λ ( T e ) , (28)where Λ ( T e ) is the radiative cooling curve taken from the CHIANTI v7.1 database(Landi et al 2013, and references therein). The energy exchange rate betweenions and electrons due to Coulomb collisions is defined in terms of the collisionfrequency ν ei N i = 2 √ m e Λ C ( e /ε ) m p (2 πk B T e ) / , (29)where m e and e are the electron mass and charge, m p is the proton mass, ε isthe vacuum permittivity, b = B /B , and Λ C is the Coulomb logarithm. Finally,the electron heat flux q e is expressed in the collisional formulation of Spitzer andH¨arm (1953): q e = κ (cid:107) bb · ∇ T e , κ (cid:107) = 3 . πΛ C (cid:114) πm e ε e ( k B T e ) / k B . (30)5.3 Transport and dissipation of Alfv´en wave turbulenceDescribing the dynamics of Alfv´en wave turbulence and its interaction with thebackground plasma requires special consideration. The evolution of the Alfv´enwave amplitude (velocity, δ u , and magnetic field, δ B ) is usually treated in termsof the Els¨asser (1950) variables, z ± = δ u ∓ δ B / √ µ ρ . The Wentzel–Kramers–Brillouin (WKB) approximation (Wentzel 1926; Kramers 1926; Brillouin 1926) isused to derive the equations that govern transport of Alfv´en waves, which may bereformulated in terms of the wave energy densities, w ± = ρ z ± /
4. Dissipation ofAlfv´en waves, Γ ± w ± , is the physical process that drives the solar wind and heatsthe coronal plasma.Alfv´en wave dissipation occurs when two counter-propagating waves interact.Alfv´en wave reflection from steep density gradients is the physical process thatresults in local wave reflection, thus maintaining a source of both types of waves.In order to describe this wave reflection we need to go beyond the WKB approx-imation that assumes that the wavelength is much smaller than spatial scales of the background variations.The equation describing the propagation, dissipation, and reflection of Alfv´enturbulence has been derived in van der Holst et al (2014): ∂w ± ∂t + ∇ · [( u ± V A ) w ± ] + w ± ∇ · u ) = − Γ ± w ± ∓ R√ w − w + , (31) where V A = B / √ µ ρ is the Alfv´en velocity, while the dissipation rate ( Γ ± ) andthe reflection coefficient ( R ) are given by Γ ± = 2 L ⊥ (cid:114) w ∓ ρ (32)and R = min (cid:26)(cid:113) ( b · [ ∇ × u ]) + [( V A · ∇ ) log V A ] , max( Γ ± ) (cid:27) ×× (cid:34) max (cid:32) − I max (cid:112) w + /w − , (cid:33) − max (cid:32) − I max (cid:112) w − /w + , (cid:33)(cid:35) . (33)Here L ⊥ is the transverse correlation length of Alfv´en waves in the plane perpendic-ular to the magnetic field line and I max = 2 is the maximum degree of turbulence“imbalance.” If (cid:112) w ± /w ∓ < I max , then Alfv´en wave reflection is neglected and R = 0.With the help of the dissipation rate of Alfv´en turbulence one can express theion and electron heating rates: Q i = f p ( Γ − w − + Γ + w + ) , Q e = (1 − f p ) ( Γ − w − + Γ + w + ) , (34)where f p ≈ . Π A : Π A B = Π A ( R (cid:12) ) B ( R (cid:12) ) = const ≈ . × (cid:20) Wm T (cid:21) . (35)The transverse correlation length is assumed to scale with the magnetic field mag-nitude (e.g., Hollweg 1986): L ⊥ ∼ B − / , (cid:104) km · T / (cid:105) ≤ L ⊥ √ B ≤ (cid:104) km · T / (cid:105) . (36)5.4 Modeling the transition region A simulation model based on the Alfv´en wave turbulence may be extrapolateddown to the top of the chromosphere. In order to save computational resources forthis physics-based model (which would require a gigantic amount of resources),the temperature and density at the top of the chromosphere are specified as: T ch = (2 ≈ × K , N ch ≈ × m − . (37)One needs to use an innovative approach to handle the sharp density gradientsthat have a spatial scale length of L = k B T ch m i g ≈ T ch × (30 m / K) , (38) xtended MHD modeling of the steady solar corona and the solar wind 37 which would greatly complicate the Alfv´en wave turbulence model and introduceunmanageable wave reflection. To avoid this problem, we apply WKB Alfv´en waveturbulence effects and let the Alfv´en waves freely propagate through the plasmaat T ≤ T ch . To both balance the radiative cooling and ensure the hydrostaticequilibrium, we apply an exponential heating function, Q h = A exp( − x/L ), tomaintain the analytical solution of the momentum and heat transfer equations, asfollows: T e = T i = T ch , N e = N i = N ch exp (cid:18) − m i gxk B ( T e + T i ) (cid:19) ,Q h = Q rad = N e Λ ( T ch ) = N ch Λ ( T ch ) exp (cid:18) − m i gxk B T ch (cid:19) . (39)Here g = 274 m / s is the gravity acceleration near the solar surface, the directionof this acceleration being antiparallel to the x-axis, and m i is the proton mass.The two constants in the solution, N ch and T ch , which are the boundary valuesfor the density and temperature, respectively, are unambiguously related to theamplitude, A , of the heating function: A = N ch Λ ( T ch ) , (40) and to the scale-length (see Eq. 38). Notice that there is a very simple relation- ship for the exponential scale-length for the heating function, which is half of thebarometric scale-length of density variation: 2 L = L g = k B ( T e + T i ) / ( m i g ).The solution satisfies the equation for the heat conduction as long as the heattransfer in the isothermic solution is absent and heating at each point exactlybalances the radiation cooling. The hydrostatic equilibrium is also maintained, aslong as k B ∂ ( N e T e + N i T i ) ∂x = − gN i m i . (41)The suggested solution does a good job describing the chromosphere. Theshort scale-length of the heating function, (see Eq. 38), which is equal to ≈ . T ch = 2 × , may presumably mimic absorption of (magneto)acousticturbulent waves, rapidly damping due to the wave-breaking effects. Physically,including this chromosphere heating function would imply that the temperaturein the chromosphere is elevated compared to the photospheric temperatures dueto some mechanism acting in the chromosphere itself. By no means can this energybe transported from the solar corona as long as the electron heat conduction rateat chromospheric temperatures is very low. One of the first successful models describing the Transition Region (TR), based onan analytical solution, was published by Lionello et al (2001a) (see also Lionello et al 2009b; Downs et al 2010). This model treats the TR as a thin continuous layerand it does not agree well with observations. This discrepancy suggests that theTR could be more accurately described as a carpet of 1D-like “threads” (maybespicules, see Cranmer et al 2013). Collectively these threads give the impressionof a “thin” layer described by a solely height-dependent 1D solution. To derive this solution one uses 1D governing equations to close the MHD model with theboundary condition at “low boundary”, which is at the same time the top boundaryfor the TR model. By solving the said 1D equations, one can merge the MHD modelto the chromosphere, which is the bottom of the TR.The heat transfer equation for a steady state hydrogen plasma in a uniformmagnetic field reads: ∂∂s (cid:18) κ T / e ∂T e ∂s (cid:19) + Q h − N e Λ ( T e ) = 0 . (42)Here Q h = Γ ( w − + w + ) is the coronal heating function, assumed to be constantat spatial scales typical for the TR. Note that the coordinate is taken along themagnetic field line, not along the radial direction.By multiplying Eq. (42) by κ T / e ( ∂T e /∂s ), and by integrating from the in-terface to the chromosphere at temperature T e , one can obtain:[ 12 κ T e (cid:18) ∂T e ∂s (cid:19) + 27 κ Q h T / e ] | T e T ch = ( N e T e ) (cid:90) T e T ch κ T / Λ ( T ) dT . (43)Here the product, N e T e , is assumed to be constant. Therefore, it is separated fromthe integrand. For a given T ch , the only parameter in the solution is ( N e T e ). Itcan be expressed at any point in terms of the local value of the heating flux andthe radiation loss integral:( N e T e ) = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) κ T e (cid:0) ∂T e ∂s (cid:1) + (cid:16) κ Q h T / e − κ Q h T / ch (cid:17)(cid:82) T e T ch κ T / Λ ( T ) dT . (44)The assumption of constant ( N e T e ) is fulfilled only if the effect of gravity isnegligible. Quantitatively, this condition is not trivial, as long as both the baro-metric scale and especially the heat conduction scale are functions of temperature.The barometric scale may be approximated as L g ( T e ) ≈ T e × (60 m / K). The heatconduction scale, L h , can be estimated by noticing that within a large part of thetransition region the radiation losses dominate over the heating function, so theyare balanced by heat conduction: κ T / e × ( T e/L h ) ∼ Q r . Thus, the condition forneglecting gravity is: L g ( T e ) ≈ T e · (60 m / K) (cid:29) L h ≈ (cid:115) κ T / e Λ ( T e )( N e T e ) . (45)In Fig. 21 we plot temperature dependencies L h ( T e ) and L g ( T e ) for ( N e T e ) =10 K / m . We see that near the chromosphere boundary the approximation givenby Eq. (45) works very well as long as the temperature changes with height arevery abrupt. The increase in temperature to 10 K occurs in less than 0.1 Mm. This estimate agrees with the temperature profile seen in the chromospheric lines (see,e.g., Fig. 2 and Fig. 8 in Avrett and Loeser (2008)). However, as the temperatureincreases with height, the effect of gravity on the temperature and density profilesbecomes more significant. It becomes comparable to the heat conduction effect at T e ≈ . × K, which can be accepted as the coronal base temperature, so thatthe transition region corresponds to the temperature range from T ch ≈ × K xtended MHD modeling of the steady solar corona and the solar wind 39 R sun Electron temperature, [K] S p a t i a l s c a l e s ( L h , L g ) , [ m ] Fig. 21
Typical scales of the transition region: the heat conduction scale (blue), L h , and thegravitational height (red), L g . to 4 . × K, with a typical width of ∼
10 Mm ≈ R S /
70. The TR solution mergesto the chromosphere solution with no jump in pressure. The merging point in thechromosphere, therefore, is at the density of ( N e T e ) /T ch ∼ cm − . The shortheat conduction scale at the chromosphere temperature (see Fig. 21) ensures thatthe heat flux from the solar corona across the transition region does not penetrateto higher densities.In Section 7 we revisit the transition region analytical model. However, in Sec-tion 6 we use another way to model the transition region, by artificially increasingthe heat conduction in the lower temperature range (see Abbett 2007). Considerthe transformation of the temperature functions shown in Eqs.(42–43): κ → f κ , ds → f ds, Γ → Γ/f, Q rad → Q rad /f, (46)with a common factor, f ≥
1. The equations do not change in this transformationand the only effect on the solution is that the temperature profile in the transitionregion becomes a factor of f wider. By applying the factor, f = ( T m /T e ) / at T ch ≤ T e ≤ T m , the heat conduction scale in this range is almost constant and isclose to ≈ T m ≈ . × K (see Fig. 21).It should be emphasized, however, that the choice of temperature range for thistransformation is highly confined by the condition given in Eq. (45). If a highervalue of T m is chosen, the heat conduction scale at the chromospheric tempera- ture exceeds the barometric scale in the chromosphere, resulting in a unphysicalpenetration of the coronal heat into the deeper chromosphere. The global model ofthe solar corona, with this unphysical energy sink, suffers from reduced values forthe coronal temperature and produces a visible distortion in the EUV and X-raysynthetic images. Thus, in formulating the transition region model we modify the heat conduction, the radiation loss rate, and the wave dissipation rate, and themaximum temperature for this modification does not exceed T m ≈ . × K. in-situ exploration of the interplanetary medium. Alfv´en waves have long beenmeasured in situ in the solar wind (Belcher and Davis, Jr. 1971), and have morerecently been remotely observed in the solar corona (De Pontieu et al 2007; Cran-mer et al 2009), where their energy is sufficient to heat and accelerate the solarwind. The theoretical exploration of Alfv´en waves was first suggested in earlywork by Hollweg (1978, 1981); Hollweg et al (1982). Based on this early work,theories were developed that describe the evolution and transport of Alfv´enic tur-bulence, e.g.,Zank et al (1996); Matthaeus et al (1999); Zank (2014); Zank et al(2017). To self-consistently describe the heating and acceleration of the solar windwith Alfv´enic turbulence, several extended magnetohydrodynamic (MHD) mod-els have been developed. One-dimensional models include those by e.g.,Tu andMarsch (1997); Laitinen et al (2003); Vainio et al (2003); Suzuki (2006); Cranmerand Van Ballegooijen (2010); Adhikari et al (2016), while multi-dimensional mod-els include e.g.,Usmanov et al (2000); Suzuki and Inutsuka (2005); Cranmer et al(2009); van der Holst et al (2010); Lionello et al (2014a).These models have many common features. First, they employ low-frequencyAlfv´en waves, which are assumed to dissipate below the ion cyclotron frequency.Wave amplitudes are typically prescribed at the inner boundaries to match ob-served wave motions in the low corona (De Pontieu et al 2007). Wave energypropagates at the Alfv´en speed along the magnetic field and drives the corona intwo ways: (i) wave pressure gradient provides a volumetric force that acceleratesthe solar wind, while (ii) wave dissipation heats the plasma. In a variety of imple-mentations, these Alfv´en wave-driven models have been shown to self-consistentlyreproduce the fast/slow solar wind speed distribution (e.g., Usmanov et al 2000;van der Holst et al 2010, 2014).An alternative method of coronal heating was developed by (Suzuki 2002,2004). Here, the focus is on shock wave heating instead of turbulent dissipation.The assumption is that slow and fast magneto-acoustic waves are generated bysmall scale reconnection events. These wave steepen into shocks while propagatingalong the field lines into the corona to heat and accelerate the plasma. Suzuki(2004) demonstrated that the dissipation of shock trains can satisfactory reproducethe fast and slow wind speeds, except for the observed high temperatures in the slow wind, where other heating mechanisms might be needed. To the best of ourknowledge, there are no 3D simulations that self-consistently include shocks andturbulence to assess which mechanism dominates in the heating and acceleration.The recently developed Alfv´en Wave Solar Model (AWSoM) (Sokolov et al2013; van der Holst et al 2014; Meng et al 2015) extends the description with xtended MHD modeling of the steady solar corona and the solar wind 41 a three-dimensional solar corona/solar wind model that self-consistently incorpo-rates low-frequency Alfv´en wave turbulence. The model employs a phenomenolog-ical treatment of wave dissipation, with a prescribed correlation length inverselyproportional to the magnetic field strength. In this case, the wave spectrum isnot resolved, so only the total forward and backward propagating wave energydensities and the partitioning of dissipated wave energy between electrons andprotons is fixed. Turbulence parameters are the wave energy densities, the corre-lation length, and the reflection rate. The wave reflection model used is essentiallythe same formulation as introduced by Matthaeus et al (1999). In this case, theenergy of the dominant wave is transferred to a counter-propagating minor wave,with the reflection coefficient controlled by the gradient of the Alfv´en speed.With AWSoM, Alfv´en waves are represented as two discrete populations prop-agating parallel and antiparallel to the magnetic field, which are imposed at theinner boundary with a Poynting flux of the outbound Alfv´en waves assumed to beproportional to the magnetic field strength. The wave spectrum is not resolved, sothe waves are presented as frequency-integrated wave energies that propagate par-allel to the magnetic field at the local Alfv´en speed. The waves possess a pressurethat does work and drives the expansion of the plasma. In this model, outwardpropagating waves experience partial reflection on field-aligned Alfv´en speed gra-dients and the vorticity of the background. The partial reflection leads to nonlinearinteraction between oppositely propagating Alfv´en waves and results in an energycascade from the large outer scale through the inertial range to the smaller per-pendicular gyroradius scales, where the dissipation takes place. The apportioningof the dissipated wave energy to the isotropic electron temperature and the paral-lel and perpendicular proton temperatures depends on criteria for the particularkinetic instabilities that are involved (Meng et al 2015). To apportion heating tothe various ion species, we use the multispecies generalization of the stochasticheating, as described by Chandran et al (2013).In the AWSoM model, the partitioning strategy is based on the dissipationof kinetic Alfv´en waves (KAWs) with the stochastic heating mechanism for theperpendicular proton temperature (Chandran et al 2011). In this mechanism, theelectric field fluctuations due to perpendicular turbulent cascade can disturb theproton gyro motion enough to give rise to perpendicular stochastic heating, assum-ing that the velocity perturbation at the proton gyroradius scale is large enough.The firehose, mirror, and ion-cyclotron instabilities, due to the developing protontemperature anisotropy, are accounted for. When the plasma is unstable becauseof these instabilities, the parallel and perpendicular temperatures are relaxed backtoward marginal stable temperatures, with the relaxation time inversely propor-tional to the growth rate of these instabilities. See the work of Meng et al (2012,2015) for a detailed description. In this global model, excess of energy in the lowercorona is transported back to the upper chromosphere via electron heat conductionwhere it is lost via radiative cooling.The AWSoM model is representative of the state of the art of extended MHDAlfv´en wave-driven coronal models, presenting many significant advances in model- ing capability. First, turbulent dissipation rates are based directly on counter prop-agating wave amplitudes, which are greatly enhanced by wave reflection at Alfv´enspeed gradients. Second, the model captures temperature anisotropies caused bypreferential perpendicular heating in the fast solar wind. Third, the effects of ki-netic instabilities: fire hose, mirror mode, and cyclotron instabilities limit temper- ature anisotropies with thresholds that are dependent on the proton temperatureratio and plasma β . Finally the three-dimensional model includes the entire struc-ture of the corona including active regions and slow and fast streams. This is thefirst time such kinetic physics has been incorporated into a global numerical modelof a CME propagating through the solar corona, which allows us to address bothparticle heating, Alfv´en wave damping, and their nonlinear coupled interaction asshown in Manchester and Van Der Holst (2017).6.2 Multi-temperature coronal modelsMagnetohydrodynamic (MHD) theory is the simplest self-consistent model de-scribing the macroscopic structure of the corona comprising the global distribu-tion and temperature of the coronal plasma and magnetic field. Such MHD modelsignore the extreme complexity of a coronal environment that is made up of manyplasma species affected by a wide range of wave-particle and particle-particle inter-actions, where heating occurs by the dissipation of waves and time varying electriccurrents. Particle populations are far from equilibrium and exhibit vastly differenttemperatures and distribution functions with extended high energy tails, the fullcomplexity of which can only be described by kinetic models with non-Maxwellianvelocity distribution functions (Landi and Pantellini 2003). For electrons, there aretwo nearly isotropic populations: the thermal core and the suprathermal halo, anda field aligned strahl component (Rosenbauer et al 1977) that travels away fromthe Sun. Ions are more often characterized by a population that is anisotropic witha temperature perpendicular to the magnetic field higher than that parallel to thefield. Hydrogen is fully ionized, and all other atomic species are highly ionized.Protons, being almost 2000 times more massive than electrons, thermodynami-cally decouple at a distance of roughly 1.5 solar radii where Coulomb collisionsbecome infrequent.To begin to address a range of physical processes as well as reduce the number offree parameters and ad hoc assumptions, a new generation of extended MHD globalcoronal models were developed. First and foremost, thermodynamic processes wereadded, beginning with heat conduction and radiative losses, which allowed modelsto accurately capture the temperature structure of the lower atmosphere. The useof stretched radial grids allow these models to resolve the transition region so thatthey may extend down to the upper chromosphere (Lionello et al 2009a; Downset al 2010; Sokolov et al 2013; van der Holst et al 2014). The radiative looses forthese models are almost universally based on CHIANTI tables (Dere et al 1997),which specify the optically thin losses from the corona, which is dominated by lineemission from heavy ions impacted by thermal electrons.With the thermal processes captured, extended MHD simulations can success-fully reproduce images of the low corona provided by extreme ultraviolet imagingtelescopes, including SOHO/EIT, STEREO/EUVI, and SDO/AIA. Observationsprovided by Downs et al (2010) and van der Holst et al (2014) indicate the qual- itative match to coronal temperature and density available with the new models.Figure 19 provides an example of coronal ultraviolet images from Jin et al (2017).Here, the simulated active regions for 7 March 2011 (CR2107) naturally producethe enhanced emissions observed by SDO/AIA 211˚A, STEREO A/EUVI 171˚A,and STEREO B/EUVI 195˚A. The distinct feature in the present model is the xtended MHD modeling of the steady solar corona and the solar wind 43 enhanced wave reflection in the presence of strong magnetic fields, such as in closeproximity to active regions that can increase the dissipation and thereby intensifythe observable EUV emission. Fig. 22
Temperature and heating rates for the three-temperature steady-state solar windsolution adapted from van der Holst et al (2014). Left panels show (top to bottom, respectively)color images of perpendicular and parallel proton temperatures and electron temperatures.Magnetic field lines are shown, ignoring the out-of-plane component. Right panels show (topto bottom, respectively) the fractions of perpendicular and parallel proton heating and electronheating by turbulent dissipation.
Even more complex, nonequilibrium thermodynamics can be captured withmultiple-temperature single-fluid coronal models. Two-temperature coronal mod-els describe protons and electrons with a single fluid velocity but with individual energy equations and temperatures (e.g., Sturrock and Hartle 1966; van der Holstet al 2010). The impetus for the feature stems from two facts: First, protons are al-most 2000 times more massive than electrons, so that at one million degrees, theirrespective sound speeds are 120 km/s and 5000 km/s. Second, Coulomb collisionsare so infrequent that within a fraction of a solar radius above the surface theions and electrons thermally decouple from each other. Consequently, heat con-duction in the corona is completely dominated by electrons, which is particularlyconspicuous in CME-driven shocks.The speed of fast CMEs occurs in the range where protons are shocked but notthe electrons, and beyond a distance of two solar radii ( R (cid:12) ) collisions become soinfrequent that protons and electrons thermally decouple on the timescale of theshock passage. As a result, protons can be shock heated to high temperature, whilein the same location electrons cool from adiabatic expansion and heat conduction.These temperature structures in CMEs were first modeled with one-dimensionaltwo-temperature simulations by Kosovichev and Stepanova (1991) and Stepanovaand Kosovichev (2000), and later in three-dimensional simulations by Manchesteret al (2012) and Jin et al (2013).The three-temperature thermodynamics model captures the electron temper-ature and resolves proton temperature into components parallel and perpendic-ular to the magnetic field. Such models can capture the temperature anisotropyproduced by a nearly collisionless plasma heated by wave-particle interaction. Aleading example is the three-temperature version of AWSoM described in van derHolst et al (2014) and Meng et al (2015) and shown in Fig. 22. Here, particletemperatures and heating rates are shown on the left and right, respectively, andare determined by the incorporation of a description of turbulent dissipation de-veloped by Chandran et al (2013). This theory describes the turbulent cascadeand dissipation of kinetic Alfv´en waves, providing the thermal energy partitioningbetween protons and electrons. As seen in Fig. 22, for regions of low plasma beta,such as coronal holes, most energy goes to perpendicular proton heating, while inhigh beta regions, such as the current sheet, parallel heating dominates. Electronheating dominates at intermediate beta levels found at the margins of the currentsheet. In AWSoM, temperature anisotropies are limited by kinetic instabilities,which are invoked when temperature ratios surpass the instability thresholds offire hose, mirror mode, and cyclotron kinetic instabilities. This three-temperaturemodel has also been applied to study the thermodynamics and the interactionAlfv´en turbulence with CMEs and CME-driven shocks (Manchester and Van DerHolst 2017). magnitude over ∼ km, resulting in a temperature gradient of ∼ K/km. Toresolve this gradient 3-D numerical simulations require sub-kilometer grid spacing,making these simulations computationally very expensive.An alternative approach is to reformulate the mathematical problem in theregion between the chromosphere and the corona in a way that decreases the xtended MHD modeling of the steady solar corona and the solar wind 45 computational cost. Instead of solving a computationally expensive 3-D problemon a very fine grid, one can reformulate it in terms of a multitude of much simpler1-D problems along threads that allows us to map the boundary conditions fromthe the solar surface to the corona. This approach is called the
Threaded-Field-LineModel (TFLM) (Sokolov et al 2016).The physics behind the reformulated problem is the assumption that betweenthe solar surface and the top of the transition region ( R (cid:12) < r < R b ) the magneticfield can be described with a scalar potential. A thread represents a field line ofthis potential field. One can introduce a 1-D problem that describes a magneticflux tube around a given thread Sokolov et al (2016).The magnetic field is divergenceless, therefore the magnetic flux remains con-stant along each thread: B ( s ) · A ( s ) = const , (47)where s is the distance along the field line, and A ( s ) is the cross-section area of theflux tube. Other conservation laws are also greatly simplified due to the fact thatin a low-beta plasma, the flow velocity is aligned with the magnetic field. Assumingsteady-state, the basic conservation laws can be written as 1-D equations.Continuity equation: ∂∂s (cid:16) ρuB (cid:17) = 0 ⇒ (cid:16) ρuB (cid:17) = const . (48)Conservation of momentum: ∂p∂s = − b r GM (cid:12) ρr ⇒ p = p TR exp r (cid:90) R TR GM (cid:12) m p k B T ( r (cid:48) ) d (cid:18) r (cid:48) (cid:19) , (49)here p = p i + p e , 2 T = T i + T e , R TR is the radius of the bottom of the transitionregion (TR), and b r is the radial component of b . In this expression terms propor-tional to u are neglected, j × B is omitted due to the fact that electric currentsvanish in a potential field ( j ∝ ∇ × B = 0), and the pressure of Alfv´en waveturbulence is assumed to be much smaller than the thermal pressure, p A (cid:28) p .Conservation of energy:2 N i k B B ( γ − ∂T∂t + 2 k B γγ − (cid:18) N i uB (cid:19) ∂T∂s = ∂∂s (cid:18) κ (cid:107) B ∂T∂s (cid:19) + Γ − w − + Γ + w + − N e N i Λ ( T ) B + (cid:16) ρuB (cid:17) ∂∂s (cid:18) GM (cid:12) r (cid:19) , (50)where the term ∂T /∂t is retained because it is assumed that the electron heatconduction is a relatively slow process.In addition to the plasma equations, the Alfv´en wave dynamics can also bereformulated. In Eq. (31), we introduce a new variable, a ± : a ± = V A Π w ± . (51)With the help of this substitution, the Alfv´en wave transport equation becomes ∂a ± ∂t + ∇ · (cid:16) u a ± (cid:17) ± ( V A · ∇ ) a ± = ∓R a − a + − (cid:118)(cid:117)(cid:117)(cid:116) Π B µ V A (cid:16) L ⊥ √ B (cid:17) a ± a ∓ . (52) These equations can be additionally simplified since in the lower corona u (cid:28) V A (i.e., waves are assumed to travel fast and quickly converge to equilibrium),therefore we can neglect the ∂a ± /∂t terms: ± ( b · ∇ ) a ± = ∓ R V A a − a + − (cid:118)(cid:117)(cid:117)(cid:116) Π B µ V A (cid:16) L ⊥ √ B (cid:17) a ± a ∓ . (53)Additionally, we introduce a new variable: dξ = ds (cid:118)(cid:117)(cid:117)(cid:116) Π B µ V A (cid:16) L ⊥ √ B (cid:17) . (54)Now the wave equations become ± da ± dξ = ∓ dsdξ R V A a ∓ − a − a + . (55)Equations (55) describe boundary value problems and one needs to specifyboundary conditions somewhere along the thread. Let ξ − denote the location ofthe lower boundary at the outgoing end of the thread (where the field directionpoints away from the Sun), and ξ + denote the lower boundary at the downwardend of the thread. The boundary conditions now must specify the values of a ± atthe location where the Alfv´en turbulence enters into the thread: a + ( ξ = ξ − ) = a + and a − ( ξ = ξ + ) = a − . The values of a ± are empirically specified.7.2 From the transition region to the threaded field line coronaFinally, one must specify the plasma and turbulence conditions at the interfacebetween the threaded field line region and the corona at the radial distance of r = R b . These conditions depend on the direction of the magnetic field at theinterface.If the magnetic field points outward, b r ( r = R b ) > (cid:16) uB (cid:17) T F = (cid:18) u · B B (cid:19) cor ; ( a − ) T F = ( a − ) cor ; ( a + ) cor = ( a + ) T F . (56)If the magnetic field points inward, b r ( r = R b ) < (cid:16) uB (cid:17) T F = − (cid:18) u · B B (cid:19) cor ; ( a + ) T F = ( a + ) cor ; ( a − ) cor = ( a − ) T F . (57) In addition, the temperature and density need to be matched at the interfacebetween the threaded field line and the corona. In order to achieve this it is assumedthat at the interface the temperature gradient is mainly in the radial direction: (cid:18) ∂T∂r (cid:19) cor = 1 b r (cid:18) ∂T∂s (cid:19) T F . (58) xtended MHD modeling of the steady solar corona and the solar wind 47 The boundary condition for the density is controlled by the sign of b · u :for b · u > (cid:18) N i uB (cid:19) T F = ( N i ) T F (cid:16) uB (cid:17) cor ;for b · u < (cid:18) N i uB (cid:19) T F = (cid:18) N i uB (cid:19) cor . (59)In the last step, one needs to consider the energy balance in the transitionregion where two physical processes balance each other: heat conduction and ra-diative cooling. Assuming steady-state conditions, this energy balance can be ex-pressed using Eq. (42) for Q h = 0, ∂∂s (cid:18) κ T / ∂T∂s (cid:19) = N e N i Λ ( T ) , (60)where for the field-aligned heat conduction coefficient the usual κ (cid:107) = κ T / ex-pression is used. The length of a given magnetic field line between the photosphereand the top of the transition region is obtained by integrating the thread: L T R = R TR (cid:90) R (cid:12) ds . (61)If the temperature at the top of the transition region, T T R , is known, one canobtain the heat flux and pressure from the following equations: N i k B T = 1 L T R (cid:90) T TR T ch κ ˜ T / d ˜ TU heat ( ˜ T ) , (62)where T ch ≈ (1 ÷ × K and κ T / T R (cid:18) ∂T∂s (cid:19) T = T TR = N i k B T U heat ( T T R ) . (63)Here U heat ( T ) = (cid:115) k B (cid:90) TT ch κ ( T (cid:48) ) / Λ ( T (cid:48) ) dT (cid:48) . (64)The Λ ( T ) and U heat ( T ) functions can be easily tabulated using the CHIANTIdatabase (Dere et al 1997; Landi et al 2013). This review gives an historical introduction to large-scale modeling of the near-steady solar corona and the solar wind. It focuses on the “quiet” corona whenthe global structure is time-independent in the frame of reference corotating withthe Sun. We start with an extensive – and critical – review of the early conceptsof the solar corona and the Biermann (1951)-Chapman (1957) puzzle that led to
Parker’s (1958) revolutionary – and highly controversial – idea of the continuoussolar wind.Following the evolution of the solar wind concept, we describe the first numer-ical models of the expanding solar corona (Noble and Scarf 1963; Scarf and Noble1965), and the emergence of the potential magnetic field model (Schatten 1968,1969) and source surface model (Altschuler and Newkirk 1969; Schatten 1969).Since the 1990s, significant progress has been made in describing and modelingthe heliosphere. MHD turned out to be surprisingly successful in describing thesolar wind from tens to hundreds of R (cid:12) (cf. Pizzo 1991; Odstrˇcil and Pizzo 1999;Groth et al 1999a; Linde et al 1998; Pogorelov et al 2013; Opher et al 2003).However, modeling the transition from the dense, cold photosphere to the hotsuper-Alfv´enic corona is still challenging. The first generation of coronal modelsused simplified energetics (cf. Usmanov 1993; Miki´c et al 1999; Cohen et al 2007)with considerable success. “Thermodynamic” models use a realistic adiabatic indexbut empirical heating functions to heat and accelerate the solar wind (cf. Grothet al 1999b; Lionello et al 2001b). The advantage of the thermodynamic approachis their ability to describe shock related phenomena.The latest generation of coronal models use Alfv´en waves to heat and acceleratethe solar wind (cf. Usmanov et al 2000; van der Holst et al 2014). While thisapproach has the promise to explain the origin of both the fast and slow solarwind states, it is still at a relatively early state of development and much workremains to be done. We urge the reader to stay tuned.Though model validation goes beyond the scope of this paper, one can en-vision investigations to determine the veracity of the models described in thisreview. As it was pointed out, one can compare emission properties in the EUVand soft x-ray with observed signatures in the low corona. In the extended corona,many of Alfv´enic effects may be more pronounced, such as non-thermal velocitiesand signatures of wave dissipation such as temperature anisotropies. The ParkerSolar Probe is ideally designed to resolve these questions with high cadence ob-servations of electromagnetic waves and particle distribution functions. One canalso validate models by comparing their predictions of solar wind parameters withthose from the WSA model. In addition, efforts are now underway to develop anintegrated model describing the acceleration and transport of solar energetic par-ticles (SEPs) directly coupled with an Alfw´en wind turbulence based solar windmodel (see Borovikov et al 2018). Such a coupled model addresses the effects ofwave turbulence on SEP transport. In particular, particle diffusion rates, and ar-rival times, are affected by the turbulence level in the solar corona. Therefore, SEPobservations may provide additional validation capability for the coronal model. Acknowledgements
The work performed at the University of Michigan was partially sup-ported by the National Science Foundation grants AGS-1322543 and PHY-1513379, NASAgrant NNX13AG25G, the European Union’s Horizon 2020 research and innovation programunder grant agreement No 637302 PROGRESS. We would also like to acknowledge high-performance computing support from: (1) Yellowstone (ark:/85065/d7wd3xhc), provided byNCAR’s Computational and Information Systems Laboratory, sponsored by the National Sci-ence Foundation, and (2) Pleiades, operated by NASA’s Advanced Supercomputing Division.xtended MHD modeling of the steady solar corona and the solar wind 49
References
Abbett WP (2007) The Magnetic Connection between the Convection Zone and Corona in theQuiet Sun. Astrophys J 665:1469–1488, DOI 10.1086/519788Adhikari L, Zank GP, Hunana P, Hu Q (2016) The Interaction of Turbulence with Paralleland Perpendicular Shocks: Theory and Observations at 1 au. Astrophys J 833:218, DOI10.3847/1538-4357/833/2/218Akasofu SI (1995) A note on the Chapman-Ferraro theory. In: Song P, Sonnerup B, ThomsenM (eds) Physics of the Magnetopause, Geophys. Monog. Ser., vol 90, AGU, pp 5–8, DOI10.1029/GM090p0005Alazraki G, Couturier P (1971) Solar Wind Acceleration Caused by the Gradient of Alfv´enWave Pressure. Astron & Astrophys 13:380Alfv´en H (1942) Existence of Electromagnetic-Hydrodynamic Waves. Nature 150:405–406,DOI 10.1038/150405d0Alfv´en H (1957) On the theory of comet tails. Tellus 9(1):92–96, DOI 10.1111/j.2153-3490.1957.tb01855.xAltschuler MD, Newkirk G (1969) Magnetic Fields and the Structure of the Solar Corona. I:Methods of Calculating Coronal Fields. Sol Phys 9:131–149, DOI 10.1007/BF00145734Antiochos SK, Linker JA, Lionello R, Miki˘c Z, Titov V, Zurbuchen TH (2012) The structureand dynamics of the corona–heliosphere connection. Space Sci Rev 172:169–185, DOI10.1007/s11214-011-9795-7Arge C, Luhmann J, Odstrcil D, Schrijver C, Li Y (2004) Stream structure and coronal sourcesof the solar wind during the may 12th, 1997 cme. J Atmos Solar-Terr Phys 66(15):1295 –1309, DOI 10.1016/j.jastp.2004.03.018Arge CN, Pizzo VJ (2000) Improvement in the prediction of solar wind conditions using near-real time solar magnetic field updates. J Geophys Res 105:10,465–10,480, DOI 10.1029/1999JA900262Avrett EH, Loeser R (2008) C. Astrophys J Suppl 175:229Barnes A (1966) Collisionless Damping of Hydromagnetic Waves. Physics of Fluids 9:1483–1495, DOI 10.1063/1.1761882Barnes A (1968) Collisionless Heating of the Solar-Wind Plasma. I. Theory of the Heating ofCollisionless Plasma by Hydromagnetic Waves. Astrophys J 154:751, DOI 10.1086/149794Belcher JW, Davis, Jr L (1971) Large amplitude Alfv´en waves in the interplanetary medium.J Geophys Res 76:3534, DOI 10.1029/JA076i016p03534Belcher JW, Davis L Jr, Smith EJ (1969) Large-amplitude Alfv´en waves in the interplanetarymedium: Mariner 5. J Geophys Res 74:2302, DOI 10.1029/JA074i009p02302Biermann L (1951) Kometenschweife und Solare Korpuskularstrahlung. Zeitschrift f¨ur Astro-physik 29:274–286Birkeland K (1908) On the Cause of Magnetic Storms and the Origin of Terrestrial Magnetism,vol 1. Christiania, Aschehoug and Co.Birkeland K (1916) Are the Solar Corpuscular Rays that penetrate the Earth’s AtmosphereNegative or Positive Rays? 1, ChristianiaBonetti A, Bridge HS, Lazarus AJ, Rossi B, Scherb F (1963) Explorer 10 plasma measurements.J Geophys Res 68:4017–4063, DOI 10.1029/JZ068i013p04017Borovikov D, Sokolov IV, Roussev I, Taktakishvili A, Gombosi TI (2018) Toward QuantitativeModel for Simulation and Forecast of Solar Energetic Particles Production during GradualEvents - I: Magnetohydrodynamic Background Coupled to the SEP Model. ArXiv e-prints
Brillouin L (1926) La m´ecanique ondulatoire de schr¨odinger: une m´ethode g´en´erale de resolu-tion par approximations successives. Comptes Rendus de l’Acadmie des Sciences 183:24–26Carrington RC (1860) Description of a singular appearance seen in the Sun on September 1,1859. Mon Not Roy Astron Soc 20:13–14Chamberlain JW (1960) Interplanetary gas II. Expansion of a model solar corona. AstrophysJ 131:47–56, DOI 10.1086/146805Chamberlain JW (1961) Interplanetary Gas. III. A Hydrodynamic Model of the Corona. As-trophys J 133:675–687, DOI 10.1086/147070Chamberlain JW (1995) Supersonic solutions of the solar-wind equations. Icarus 113:450–455,DOI 10.1006/icar.1995.1034Chandran BDG, Dennis TJ, Quataert E, Bale SD (2011) Incorporating kinetic physics intoa two-fluid solar-wind model with temperature anisotropy and low-frequency Alfv´en-wave0 Tamas I. Gombosi et al.turbulence. Astrophys J 743, DOI 10.1088/0004-637X/743/2/197Chandran BDG, Verscharen D, Quataert E, Kasper JC, Isenberg PA, Bourouaine S (2013)Stochastic Heating, Differential Flow, and the Alpha-to-proton Temperature Ratio in theSolar Wind. Astrophys J 776:45, DOI 10.1088/0004-637X/776/1/45,
Chapman S (1954) The viscosity and thermal conductivity of a completely ionized gas. Astro-phys J 120:151–155, DOI { } Chapman S (1957) Notes on the solar corona and the terrestrial ionosphere. SmithsonianContributions to Astrophysics 2:1Chapman S, Ferraro VCA (1931a) A new theory of magnetic storms (Sections 1–5). Terr MagnAtmos Electr 36(2):77–97, DOI 10.1029/TE036i002p00077Chapman S, Ferraro VCA (1931b) A new theory of magnetic storms (Sections 6-7). Terr MagnAtmos Electr 36(3):171–186, DOI 10.1029/TE036i003p00171Chapman S, Ferraro VCA (1932a) A new theory of magnetic storms (Section 8). Terr MagnAtmos Electr 37(2):147–156Chapman S, Ferraro VCA (1932b) A new theory of magnetic storms (Section 9). Terr MagnAtmos Electr 37(4):421–429, DOI 10.1029/TE037i004p00421Chapman S, Ferraro VCA (1933) A new theory of magnetic storms (Sections 10-11). TerrMagn Atmos Electr 38(2):79–96, DOI 10.1029/TE038i002p00079Cohen O, Sokolov IV, Roussev II, Arge CN, Manchester WB, Gombosi TI, Frazin RA, ParkH, Butala MD, Kamalabadi F, Velli M (2007) A semiempirical magnetohydrodynamicalmodel of the solar wind. Astrophys J Lett 654:L163–L166, DOI 10.1086/511154Cohen O, Sokolov IV, Roussev II, Gombosi TI (2008) Validation of a synoptic solar windmodel. J Geophys Res 113(A3):A03,104, DOI 10.1029/2007JA012797Coleman PJ Jr (1966) Variations in the interplanetary magnetic field: Mariner 2: 1. Observedproperties. J Geophys Res 71:5509–5531, DOI 10.1029/JZ071i023p05509Coleman PJ Jr (1967) Wave-like phenomena in the interplanetary plasma: Mariner 2. PlanetSpace Sci 15:953–973, DOI 10.1016/0032-0633(67)90166-3Coleman PJ Jr (1968) Turbulence, Viscosity, and Dissipation in the Solar-Wind Plasma. As-trophys J 153:371, DOI 10.1086/149674Cranmer SR, Van Ballegooijen AA (2010) Can the solar wind be driven by magnetic reconnec-tion in the Sun’s magnetic carpet? Astrophys J 720(1):824–847, DOI 10.1088/0004-637X/720/1/824Cranmer SR, Matthaeus WH, Breech BA, Kasper JC (2009) Empirical constraints on protonand electron heating in the fast solar wind. Astrophys J 702(2):1604Cranmer SR, van Ballegooijen AA, Woolsey LN (2013) Connecting the sun’s high-resolutionmagnetic carpet to the turbulent heliosphere. The Astrophysical Journal 767(2):125, URL http://stacks.iop.org/0004-637X/767/i=2/a=125
De Pontieu B, McIntosh SW, Carlsson M, Hansteen VH, Tarbell TD, Schrijver CJ, Title AM,Shine RA, Tsuneta S, Katsukawa Y, Ichimoto K, Suematsu Y, Shimizu T, Nagata S (2007)Chromospheric alfv´enic waves strong enough to power the solar wind. Science 318:1574–,DOI 10.1126/science.1151747Del Zanna L, Velli M, Londrillo P (1998) Dynamical response of a stellar atmosphere topressure perturbations: numerical simulations. Astron Astrophys 330:L13–L16, DOI 10.1051/0004-6361:20000455Dere K, Landi E, Mason H, Monsignori-Fossi B, Young P (1997) Chianti - an atomic databasefor emission lines. Astron Astrophys Suppl Ser 125(1):149–173, DOI 10.1051/aas:1997368Dessler AJ (1967) Solar wind and interplanetary magnetic field. Rev Geophys 5:1–41, DOI10.1029/RG005i001p00001Dessler AJ (1984) The evolution of arguments regarding the existence of field-aligned currents.In: Potemra TA (ed) Magnetospheric Currents, Geophys. Monog. Ser., vol 28, AGU, Wash-ington, D. C., pp 22–28, DOI 10.1029/GM028p0022Dewar RL (1970) Interaction between Hydromagnetic Waves and a Time-Dependent, Inhomo-geneous Medium. Physics of Fluids 13:2710–2720, DOI 10.1063/1.1692854Dmitruk P, Matthaeus WH, Milano LJ, Oughton S, Zank GP, Mullan DJ (2002) Coronal Heat-ing Distribution Due to Low-Frequency, Wave-driven Turbulence. Astrophys J 575:571–577, DOI 10.1086/341188Downs C, Roussev II, van der Holst B, Lugaz N, Sokolov IV, Gombosi TI (2010) Toward a re-alistic thermodynamic magnetohydrodynamic model of the global solar corona. AstrophysJ 712(2):1219–1231, DOI 10.1088/0004-637X/712/2/1219xtended MHD modeling of the steady solar corona and the solar wind 51Edl´en B (1941) An attempt to identify the emission lines in the spectrum of the solar corona.Ark Mat Astron Fys 28B:1–4Els¨asser WM (1950) The Hydromagnetic Equations. Physical Review 79:183–183, DOI 10.1103/PhysRev.79.183Feng X, Zhou Y, Wu ST (2007) A novel numerical implementation for solar wind modeling bythe modified conservation element/solution element method. Astrophys J 655:1110–1126,DOI 10.1086/510121Feng X, Yang L, Xiang C, Wu ST, Zhou Y, Zhong D (2010) Three-dimensional Solar WINDModeling from the Sun to Earth by a SIP-/CESE MHD Model with a Six-componentGrid. Astrophys J 723:300–319, DOI 10.1088/0004-637X/723/1/300Feng X, Jiang C, Xiang C, Zhao X, Wu ST (2012a) A Data-driven Model for the GlobalCoronal Evolution. Astrophys J 758:62, DOI 10.1088/0004-637X/758/1/62Feng X, Yang L, Xiang C, Jiang C, Ma X, Wu ST, Zhong D, Zhou Y (2012b) Validationof the 3D AMR SIP-CESE Solar Wind Model for Four Carrington Rotations. solphys279:207–229, DOI 10.1007/s11207-012-9969-9Fisk LA, Schwadron NA, Zurbuchen TH (1998) On the Slow Solar Wind. Space Sci Rev86:51–60, DOI 10.1023/A:1005015527146FitzGerald GF (1892) Sunspots and magnetic storms. The Electrician 30:48Gosling JT, Bame SJ, McComas DJ, Phillips JL, Pizzo VJ, Goldstein BE, Neugebauer M(1993) Latitudinal variation of solar wind corotating stream interaction regions: Ulysses.Geophys Res Lett 20(24):2789–2792Green JL, Boardsen S (2006) Duration and extent of the great auroral storm of 1859. AdvSpace Res 38:130–135Gressl C, Veronig AM, Temmer M, Odstrˇcil D, Linker JA, Miki´c Z, Riley P (2014) ComparativeStudy of MHD Modeling of the Background Solar Wind. SOLPHYS 289:1783–1801, DOI10.1007/s11207-013-0421-6Gringauz KI, Bezrukikh VV, Balandina SM, Ozerov VD, Rybchinskiy RE (1964) Direct ob-servations of solar plasma fluxes at distances in the order of 1,900,000 km from the Earth,17 February 1961, and simultaneous observations of the geomagnetic field. Planet SpaceSci 12:87–90Gringauz KI, Bezrukikh VV, Ozerov VD, Rybchinskii RE (1960) Investigation of interplan-etary ionized gas, high energy electrons and solar corpuscular radiation by means of 3-electrode traps for charge-carrying particles installed on the 2nd Soviet space rocket. Dok-lady Akademii Nauk SSSR 131:1301–1304Groth CPT, De Zeeuw DL, Gombosi TI, Powell KG (1999a) A parallel adaptive 3D MHDscheme for modeling coronal and solar wind plasma flows. Space Sci Rev 87:193–198Groth CPT, De Zeeuw DL, Powell KG, Gombosi TI, Stout QF (1999b) A parallel solution-adaptive scheme for ideal magnetohydrodynamics. In: Proc. 14th AIAA ComputationalFluid Dynamics Conference, AIAA Paper No. 99-3273, Norfolk, VirginiaGroth CPT, De Zeeuw DL, Gombosi TI, Powell KG (2000a) Global 3D MHD simulation ofa space weather event: CME formation, interplanetary propagation, and interaction withthe magnetosphere. J Geophys Res 105(A11):25,053 – 25,078Groth CPT, De Zeeuw DL, Gombosi TI, Powell KG (2000b) Three-dimensional MHD simu-lation of coronal mass ejections. Adv Space Res 26(5):793–800Grotian W (1939) Zur Frage der Deutung der Linien im Spektrum der Sonnenkorona. Natur-wissenschaften 27:214–214, DOI 10.1007/BF01488890Hayashi K (2005) Magnetohydrodynamic Simulations of the Solar Corona and Solar WindUsing a Boundary Treatment to Limit Solar Wind Mass Flux. Astrophys J Suppl 161:480–494, DOI 10.1086/491791Hayashi K (2012) An MHD simulation model of time-dependent co-rotating solar wind. Journalof Geophysical Research (Space Physics) 117:A08105, DOI 10.1029/2011JA017490Heinemann M, Olbert S (1980) Non-WKB Alfv´en waves in the solar wind. J Geophys Res85:1311–1327, DOI 10.1029/JA085iA03p01311Hodgson R (1860) On a curious appearance seen in the Sun. Mon Not Roy Astron Soc 20:15Hoeksema JT, Wilcox JM, Scherrer PH (1982) Structure of the heliospheric current sheetin the early portion of sunspot cycle 21. J Geophys Res 87:10,331–10,338, DOI 10.1029/JA087iA12p10331Hollweg JV (1978) Alfv´en waves in the solar atmosphere. Sol Phys 56:305–333, DOI 10.1007/BF001524742 Tamas I. Gombosi et al.Hollweg JV (1981) Alfven waves in the solar atmosphere. II - Open and closed magnetic fluxtubes. Sol Phys 70:25–66, DOI 10.1007/BF00154391Hollweg JV (1986) Transition region, corona, and solar wind in coronal holes. J Geophys Res91:4111–4125, DOI 10.1029/JA091iA04p04111Hollweg JV, Jackson S, Galloway D (1982) Alfven waves in the solar atmosphere. III - Nonlinearwaves on open flux tubes. Sol Phys 75:35–61, DOI 10.1007/BF00153458van der Holst B, Sokolov IV, Meng X, Jin M, Manchester WB, T´oth G, Gombosi TI (2014)Alfv´en wave solar model (AWSoM): Coronal heating. Astrophys J 782:81, DOI 10.1088/0004-637X/782/2/81van der Holst B, Manchester WB, Frazin RA, V´asquez AM, Toth G, Gombosi TI (2010) A data-driven, two-temperature solar wind model with Alfv´en waves. Astrophys J 725(1):1373–1383, DOI 10.1088/0004-637X/725/1/1373Hu YQ, Li X, Habbal SR (2003) A 2.5-Dimensional MHD Alfv´en-Wave-Driven Solar WindModel. J Geophys Res 108:9–1, DOI 10.1029/2003JA009889Jackson BV, Hick PL, Kojima M, Yokobe A (1998) Heliospheric tomography using interplan-etary scintillation observations 1. Combined Nagoya and Cambridge data. J Geophys Res103(12):12,049–12,068Jacques SA (1977) Momentum and energy transport by waves in the solar atmosphere andsolar wind. Astrophys J 215:942–951, DOI 10.1086/155430Jacques SA (1978) Solar wind models with Alfv´en waves. Astrophys J 226:632–649, DOI10.1086/156647Jian LK, MacNeice PJ, Taktakishvili A, Odstrcil D, Jackson B, Yu HS, Riley P, Sokolov IV,Evans RM (2015) Validation for solar wind prediction at earth: Comparison of coronaland heliospheric models installed at the ccmc. Space Weather 13(5):316–338, DOI 10.1002/2015SW001174Jin M, Manchester WB, van der Holst B, Oran R, Sokolov I, Toth G, Liu Y, Sun XD,Gombosi TI (2013) Numerical simulations of coronal mass ejection on 2011 March 7:One-temperature and two-temperature model comparison. Astrophys J 773(1):50, DOI10.1088/0004-637X/773/1/50Jin M, Manchester WB, van der Holst B, Sokolov I, Tth G, Vourlidas A, de Koning CA,Gombosi TI (2017) Chromosphere to 1 au simulation of the 2011 march 7th event: Acomprehensive study of coronal mass ejection propagation. Astrophys J 834(2):172, DOI10.3847/1538-4357/834/2/172Kim TK, Pogorelov NV, Zank GP, Elliott HA, McComas DJ (2016) Modeling the solar windat the Ulysses, Voyager, and New Horizons spacecraft. Astrophys J 832(1):72, DOI 10.3847/0004-637x/832/1/72Kosovichev AG, Stepanova TV (1991) Numerical Simulation of Shocks in the Heliosphere.Soviet Astronomy 35:646Kramers HA (1926) Wellenmechanik und halbzahlige quantisierung. Zeitschrift f¨ur Physik39(10):828–840, DOI 10.1007/BF01451751Laitinen T, Fichtner H, Vainio R (2003) Toward a self-consistent treatment of the cyclotronwave heating and acceleration of the solar wind plasma. J Geophys Res 108:1081, DOI10.1029/2002JA009479Landi E, Young PR, Dere KP, Del Zanna G, Mason HE (2013) CHIANTI - An Atomic Databasefor Emission Lines. XIII. Soft X-Ray Improvements and Other Changes. Astrophys J763:86, DOI 10.1088/0004-637X/763/2/86Landi S, Pantellini F (2003) Kinetic simulations of the solar wind from the subsonic to thesupersonic regime. Astron Astrophys 400:769–778, DOI 10.1051/0004-6361:20021822Leroy B (1980) Propagation of waves in an atmosphere in the presence of a magnetic field. II- The reflection of Alfv´en waves. Astron Astrophys 91:136–146Linde TJ, Gombosi TI, Roe PL, Powell KG, De Zeeuw DL (1998) The heliosphere in themagnetized local interstellar medium: Results of a 3D MHD simulation. J Geophys Res103(A2):1889–1904, DOI 10.1029/97JA02144Linker JA, Miki´c Z, Biesecker DA, Forsyth RJ, Gibson SE, Lazarus AJ, Lecinski A, Riley P,Szabo A, Thompson BJ (1999) Magnetohydrodynamic modeling of the solar corona duringWhole Sun Month. J Geophys Res 104(A5):9809–9830Lionello R, Linker JA, Miki´c Z (2001a) Including the transition region in models of the large-scale solar corona. Astrophys J 546:542–551Lionello R, Linker JA, Miki´c Z (2001b) Including the Transition Region in Models of theLarge-Scale Solar Corona. Astrophys J 546:542–551, DOI 10.1086/318254xtended MHD modeling of the steady solar corona and the solar wind 53Lionello R, Riley P, Linker JA, Miki Z (2005) The effects of differential rotation on the magneticstructure of the solar corona: Magnetohydrodynamic simulations. Astrophys J 625(1):463–473, DOI 10.1086/429268Lionello R, Linker JA, Mikiˇc Z (2009a) Multispectral emission of the sun during the firstwhole sun month: Magnetohydrodynamic simulations. Astrophys J 690(1):902–912, DOI10.1088/0004-637X/690/1/902Lionello R, Linker JA, Miki´c Z (2009b) Multispectral emission of the sun during the first wholesun month: Magnetohydrodynamic simulations. Astrophys J 690:902–912Lionello R, Linker JA, Miki´c Z, Riley P, Titov VS (2011) MHD simulations of the globalsolar corona and the solar wind. In: Miralles MP, S´anchez-Almeida J (eds) The Sun, theSolar Wind, and the Heliosphere, Springer Netherlands, Dordrecht, pp 101–106, DOI10.1007/978-90-481-9787-3 \ { } Newton I (1718) Opticks: Or, A Treatise of the Reflections, Refractions, Inflexions and Coloursof Light., the second edition, with additions edn. unknown, LondonNoble LM, Scarf FL (1963) Conductive Heating of the Solar Wind. I. Astrophys J 138:1169–1181, DOI 10.1086/147715Norquist DC, Meeks WC (2010) A comparative verification of forecasts from two operationalsolar wind models. Space Weather 8:S12,005, DOI 10.1029/2010SW000598Odstrcil D, Riley P, Zhao X (2004) Numerical simulation of the 12 May 1997 interplanetaryCME event. JGR 109:A02,116Odstrcil D, Pizzo VJ, Arge CN (2005) Propagation of the 12 May 1997 interplanetary coronalmass ejection in evolving solar wind structures. J Geophys Res 110(A9):A02106, DOI10.1029/2004JA010745Odstrˇcil, Karlicky M (1997) Triggering of magnetic reconnection in the current sheet by shockwaves. Astron Astrophys 326:1252–1258Odstrˇcil, Dryer M, Smith Z (1996) Propagation of an interplanetary shock along the helio-spheric plasma sheet. J Geophys Res 101:19,973–19,984Odstrˇcil D (2003) Modeling 3-D solar wind structures. Adv Space Res 32(4):497–506Odstrˇcil D, Pizzo VJ (1999) Distortion of the interplanetary magnetic field by three-dimensional propagation of coronal mass ejections in a structured solar wind. J GeophysRes 104:28,225–28,239Odstrˇcil D, Pizzo VJ, Linker JA, Riley P, Lionello R, Mikic Z (2004) Initial coupling ofcoronal and heliospheric numerical magnetohydrodynamic codes. J Atmos Solar-Terr Phys66:1311–1320Odstrˇcil D, Linker JA, Lionello R, Miki´c Z, Riley P, Pizzo VJ, Luhmann JG (2002) Mergingof coronal and heliospheric numerical two-dimensional MHD models. J Geophys Res 107,DOI { } Opher M, Liewer PC, Gombosi TI, Manchester WB, De Zeeuw DL, Sokolov IV, T´oth G (2003)Probing the edge of the solar system: Formation of an unstable jet-sheet. Astrophys J591:L61–L65, DOI 10.1086/376960Opher M, Stone EC, Liewer PC, Gombosi T (2006) Global asymmetry of the heliosphere. In:AIP Conference Proceedings, AIP, DOI 10.1063/1.2359304Opher M, Drake JF, Zieger B, Swisdak M, Toth G (2016) Magnetized jets driven by the sun:The structure of the heliosphere revisited – updates. Physics of Plasmas 23(5):056,501,DOI 10.1063/1.4943526Opher M, Stone EC, Gombosi TI (2007) The orientation of the local interstellar magneticfield. Science 316(5826):875–878, DOI 10.1126/science.1139480Oran R, van der Holst B, Landi E, Jin M, Sokolov IV, Gombosi TI (2013) A global wave-drivenmagnetohydrodynamic solar model with a unified treatment of open and closed magneticfield topologies. Astrophys J 778:176–195, DOI 10.1088/0004-637X/778/2/176Owens MJ, Spence HE, McGregor S, Hughes W, Quinn JM, Arge CN, Riley P, Linker J,Odstrˇcil D (2008) Metrics for solar wind prediction models: Comparison of empirical,hybrid and physics-based schemes with 8-years of l1 observations. Space Weather 6, DOI10.1029/2007SW000380Parker EN (1958) Dynamics of the interplanetary gas and magnetic fields. Astrophys J128(3):664–676, DOI 10.1086/146579Parker EN (1963) Interplanetary Dynamical Processes. Wiley-Interscience, New YorkParker EN (2001) A history of the solar wind concept. In: Bleeker JAM, Geiss J, Huber; M(eds) The Century of Space Science, Kluwer, pp 225–255Pizzo V (1978) A three-dimensional model of corotating streams in the solar wind 1. Theoreticalfoundations. J Geophys Res 83(A12):5563–5572Pizzo VJ (1980) A three-dimensional model of corotating streams in the solar wind 2. Hydro-dynamic streams. J Geophys Res 85(A2):727–743Pizzo VJ (1982) A three-dimensional model of corotating streams in the solar wind 3. Magne-tohydrodynamic streams. J Geophys Res 87(A6):4374–4394Pizzo VJ (1989) The evolution of corotating stream fronts near the ecliptic plane in the innersolar system 1. Two-dimensional fronts. J Geophys Res 94(A7):8673–8684Pizzo VJ (1991) The evolution of corotating stream fronts near the ecliptic plane in the innersolar system 2. Three-dimensional tilted-dipole fronts. J Geophys Res 96(A4):5405–5420xtended MHD modeling of the steady solar corona and the solar wind 55Pizzo VJ (1994a) Global, quasi-steady dynamics of the distant solar wind. 1. Origin of North-South flows in the outer heliosphere. J Geophys Res 99(A3):4173–4183Pizzo VJ (1994b) Global, quasi-steady dynamics of the distant solar wind. 2. Deformation ofthe heliospheric current sheet. J Geophys Res 99(A3):4185–4191Pizzo VJ, Gosling JT (1994) 3-D simulation of high-latitude interaction regions: Comparisonwith ulysses results. Geophys Res Lett 21:2063Pizzo VJ, MacGregor KB, Kunasz PB (1993) A numerical simulation of two-dimensional ra-diative equilibrium in magnetostatic flux tubes. I - The model. Astrophys J 404:788–798,DOI 10.1086/172333Pneuman GW (1966) Interaction of the solar wind with a large-scale solar magnetic field.Astrophys J 145:242–254, DOI 10.1086/148759Pneuman GW (1968) Some general properties of helmeted coronal structures. Sol Phys pp578–597, DOI 10.1007/BF00151939Pneuman GW (1969) Coronal streamers. ii: Open streamer configurations. Sol Phys 6:255–275,DOI 10.1007/BF00150951Pneuman GW, Kopp RA (1971) Gas-magnetic field interactions in the solar corona. Sol Phys18:258–270Pogorelov NV, Suess ST, Borovikov SN, Ebert RW, McComas DJ, Zank GP (2013) Three-dimensional features of the outer heliosphere due to coupling between the interstellarand interplanetary magnetic fields. iv. solar cycle model based on ulysses observations.Astrophys J 772:2, DOI 10.1088/0004-637X/772/1/2Pogorelov NV, Borovikov SN, Heerikhuisen J, Zhang M (2015) The heliotail. Astrophys J812:L6, DOI 10.1088/2041-8205/812/1/L6Pogorelov NV, Heerikhuisen J, Roytershteyn V, Burlaga LF, Gurnett DA, Kurth WS (2017)Three-dimensional features of the outer heliosphere due to coupling between the interstellarand heliospheric magnetic field. V. The bow wave, heliospheric boundary layer, instabili-ties, and magnetic reconnection. Astrophys J 845:9, DOI 10.3847/1538-4357/aa7d4fPomoell J, Poedts S (2018) EUHFORIA: EUropean Heliospheric FORecasting InformationAsset. J of Space Weather and Space Climate Accepted:in pressRappazzo AF, Matthaeus WH, Ruffolo D, Servidio S, Velli M (2012) Interchange reconnectionin a turbulent corona. Astrophys J Lett 758(1):L14, DOI 10.1088/2041-8205/758/1/L14Reiss MA, Temmer M, Veronig AM, Nikolic L, Vennerstrom S, Sch¨ongassner F, Hofmeister SJ(2016) Verification of high-speed solar wind stream forecasts using operational solar windmodels. Space Weather 14:495–510, DOI 10.1002/2016SW001390Riley P, Linker JA, Miki´c Z, Lionello R, Ledvina SA, Luhmann JG (2006) A comparison be-tween global solar magnetohydrodynamic and potential field source surface model results.Astrophys J 653:1510–1516, DOI 10.1086/508565Riley P, Linker JA, Arge CN (2015) On the role played by magnetic expansion factor in theprediction of solar wind speed. Space Weather 13(3):154–169, DOI 10.1002/2014SW001144R¨ontgen WC (1896) On a new kind of rays. Science 3:227–231, DOI 10.1126/science.3.59.227Rosenbauer H, Schwenn R, Marsch E, Meyer B, Miggenrieder H, Montgomery MD,Muehlhaeuser KH, Pilipp W, Voges W, Zink SM (1977) A survey on initial results ofthe HELIOS plasma experiment. Journal of Geophysics Zeitschrift Geophysik 42:561–580Roussev II, Forbes TG, Gombosi TI, Sokolov IV, DeZeeuw DL, Birn J (2003) A Three-dimensional Flux Rope Model for Coronal Mass Ejections Based on a Loss of Equilibrium.Astrophys J Lett 588:L45–L48, DOI 10.1086/375442Scarf FL, Noble LM (1965) Conductive Heating of the Solar Wind. II. The Inner Corona.Astrophys J 141:1479–1491, DOI 10.1086/148236Schatten KH (1968) Prediction of the coronal structure for the solar eclipse of September 22,1968. Nature 220:1211–1213, DOI 10.1038/2201211a0Schatten KH (1969) Coronal structure at the solar eclipse of September 22, 1968. Nature222:652–653, DOI 10.1038/222652a0Schatten KH (1971) Current sheet magnetic model for the solar corona. Cosmic Electrody-namics 2:232–245Schatten KH, Wilcox JM, Ness NF (1969) A model of interplanetary and coronal magneticfields. Sol Phys 6:442–455, DOI 10.1007/BF00146478Shiota D, Kataoka R (2016) Magnetohydrodynamic simulation of interplanetary propagationof multiple coronal mass ejections with internal magnetic flux rope (SUSANOO-CME).Space Weather 14:56–75, DOI 10.1002/2015SW0013086 Tamas I. Gombosi et al.Shiota D, Kataoka R, Miyoshi Y, Hara T, Tao C, Masunaga K, Futaana Y, Terada N (2014)Inner heliosphere mhd modeling system applicable to space weather forecasting for theother planets. Space Weather 12(4):187–204, DOI 10.1002/2013SW000989Siscoe G, Odstrcil D (2008) Ways in which ICME sheaths differ from magnetosheaths. JGeophys Res 113:A00B07, DOI 10.1029/2008JA013142Sokolov IV, van der Holst B, Oran R, Downs C, Roussev II, Jin M, Manchester WB, Evans RM,Gombosi TI (2013) Magnetohydrodynamic waves and coronal heating: Unifying empiricaland MHD turbulence models. Astrophys J 764:23, DOI 10.1088/0004-637X/764/1/23Sokolov IV, Bvan der Holst, Manchester WB, Ozturk D, Szente J, Taktakishvili A, T´oth G,Jin M, Gombosi TI (2016) Threaded-field-lines model for the low solar corona powered byAlfv´en wave turbulence. arXiv:160904379 [astro-ph.SR]Spitzer L (1962) Physics of Fully Ionized Gases, 2nd edn. Interscience Publishers, Inc., NewYorkSpitzer L, H¨arm R (1953) Transport phenomena in a completely ionized gas. Phys Rev 89:977–981, DOI 10.1103/PhysRev.89.977Steinolfson RS (1988) Density and white light brightness in looplike coronal mass ejections:Importance of the pre-event atmosphere. J Geophys Res 93:14,261–14,267Steinolfson RS (1990) Coronal mass ejection shock fronts containing the two types of interme-diate shocks. J Geophys Res 95:20,693–20,699Steinolfson RS (1992) Three-dimensional structure of coronal mass ejections. J Geophys Res97:10,811–10,824Steinolfson RS (1994) Modeling coronal streamers and their eruption. Space Sci Rev 70:289–294Steinolfson RS, Hundhausen AJ (1988) Density and white light brightness in looplike coronalmass ejections: Temporal evolution. J Geophys Res 93:14,269–14,276Steinolfson RS, Dryer M, Nakagawa Y (1975) Numerical MHD simulation of interplanetaryshock pairs. J Geophys Res 80:1223–1231, DOI 10.1029/JA080i010p01223Steinolfson RS, Wu ST, Dryer M, Tandberg-Hanssen E (1978) Magnetohydrodynamic modelsof coronal transients in the meridional plane. I - The effect of the magnetic field. AstrophysJ 225:259–274, DOI 10.1086/156489Steinolfson RS, Suess ST, Wu ST (1982) The steady global corona. Astrophys J 255:730–742,DOI 10.1086/159872Stepanova T, Kosovichev A (2000) Observation of shock waves associated with coronal massejections from soho/lasco. Adv Space Res 25(9):1855 – 1858, DOI 10.1016/S0273-1177(99)00598-0Sturrock PA, Hartle RE (1966) Two-fluid model of the solar wind. Phys Rev Lett 16:628–631,DOI 10.1103/PhysRevLett.16.628Suzuki TK (2002) On the Heating of the Solar Corona and the Acceleration of the Low-SpeedSolar Wind by Acoustic Waves Generated in the Corona. Astrophys J 578:598–609, DOI10.1086/342347Suzuki TK (2004) Coronal heating and acceleration of the high/low-speed solar wind byfast/slow MHD shock trains. Mon Not Roy Ast Soc 349:1227–1239, DOI 10.1111/j.1365-2966.2004.07570.xSuzuki TK (2006) Forecasting Solar Wind Speeds. Astrophys J Lett 640:L75–L78, DOI 10.1086/503102, arXiv:astro-ph/0602062
Suzuki TK, Inutsuka Si (2005) Making the corona and the fast solar wind: A self-consistentsimulation for the low-frequency Alfv´en waves from the photosphere to 0.3 AU. AstrophysJ Lett 632:L49–L52, DOI 10.1086/497536Thomson JJ (1897) Cathode rays. The Electrician 39:104–111Thomson W (1893) President’s Address. Proc Roy Soc London 52:300–315Titov VS, Mikic Z, Linker JA, Lionello R (2008) 1997 May 12 Coronal Mass Ejection Event.I. A Simplified Model of the Preeruptive Magnetic Structure. Astrophys J 675:1614–1628,DOI 10.1086/527280,
Titov VS, Mikic Z, T¨or¨ok T, Linker JA, Panasenco O (2012) 2010 August 1-2 SympatheticEruptions. I. Magnetic Topology of the Source-surface Background Field. Astrophys J759:70, DOI 10.1088/0004-637X/759/1/70,
T¨or¨ok T, Downs C, Linker JA, Lionello R, Titov VS, Miki´c Z, Riley P, Caplan RM, Wijaya J(2018) Sun-to-earth MHD simulation of the 2000 july 14 “bastille day” eruption. AstrophysJ 856(1):75, DOI 10.3847/1538-4357/aab36dxtended MHD modeling of the steady solar corona and the solar wind 57Tu CY, Marsch E (1997) Two-Fluid Model for Heating of the Solar Corona and Acceleration ofthe Solar Wind by High-Frequency Alfven Waves. Sol Phys 171:363–391, DOI 10.1023/A:1004968327196Usmanov AV (1993) A global numerical 3-D MHD model of the solar wind. Sol Phys 146:377–396, DOI 10.1007/BF00662021Usmanov AV, Goldstein ML, Besser BP, Fritzer JM (2000) A global MHD solar wind modelwith WKB Alfv´en waves: Comparison with Ulysses data. J Geophys Res 105:12,675–12,695,DOI 10.1029/1999JA000233Vainio R, Laitinen T, Fichtner H (2003) A simple analytical expression for the power spectrumof cascading Alfv´en waves in the solar wind. Astrophys J 407:713–723, DOI 10.1051/0004-6361:20030914V´asquez AM, Frazin RA, Hayashi K, Sokolov IV, Cohen O, Manchester WB IV, Kamalabadi F(2008) Validation of Two MHD Models of the Solar Corona with Rotational Tomography.Astrophys J 682:1328–1337, DOI 10.1086/589682Velli M (1994) From supersonic winds to accretion: Comments on the stability of stellar windsand related flows. Astrophys J Lett 432:L55–L58, DOI 10.1086/187510Verdini A, Velli M (2007) Alfv´en Waves and Turbulence in the Solar Atmosphere and SolarWind. Astrophys J 662:669–676, DOI 10.1086/510710Verdini A, Velli M, Matthaeus WH, Oughton S, Dmitruk P (2010) A turbulence-driven modelfor heating and acceleration of the fast wind in coronal holes. Astrophys J Lett 708:L116–L120, DOI 10.1088/2041-8205/708/2/L116,0911.5221