Astronomy & Astrophysics manuscript no. Aibeo2006 c ESO 2008 (cid:13) February 5, 2008 Application of a MHD hybrid solar wind model with latitudinal Ulysses dependences to data at minimum 7 0 A. Aib´eo1,2, J.J.G. Lima1,3 and C. Sauty4 0 2 1 Centro deAstrof´ısica da Universidadedo Porto, Rua das Estrelas, 4150-762 Porto, Portugal n 2 Departamento de Engenharia Mecˆanica e Gest˜ao industrial da Escola Superior de Tecnologia de Viseu, Campus Polit´ecnico a deViseu, 4105 Viseu, Portugal J 3 DepartamentodeMatem´atica Aplicada daFaculdadedeCiˆencias, UniversidadedoPorto, RuadoCampo Alegre,657, 4169- 4 007 Porto, Portugal 4 Observatoire deParis, L.U.Th., 92190 Meudon,France 1 v 4 Received ****** **, 2006; accepted *** **, 2006 9 0 ABSTRACT 1 0 Aims. In a previous work, Ulysses data was analyzed to build a complete axisymmetric MHD solution for the solar wind 7 at minimum including rotation and the initial flaring of the solar wind in the low corona. This model has some problems in 0 reproducingthevaluesofmagneticfieldat1AUdespitethecorrectvaluesofthevelocity.Here,weintendtoextendtheprevious / h analysis to anothertypeof solutions and to improveour modelling of the wind from thesolar surface to 1 AU. p Methods.Wecomparethepreviousresultstothoseobtainedwithafullyhelicoidalmodelandconstructahybridmodelcombining - both previous solutions, keeping the flexibility of the parent models in the appropriate domain. From the solar surface to the o Alfv´en point, athreecomponent solution for velocity and magnetic field isused, reproducingthecomplex wind geometry and r t thewell-knownflaringofthefieldlinesobservedincoronalholes.FromtheAlfv´enradiusto1AUandfurther,thehybridmodel s a keepsthelatitudinaldependencesasflexibleaspossible, inordertodealwiththesharpvariationsneartheequatorandweuse : thehelicoidal solution, turning thepoloidal streamlines into radial ones. v Results.Despitetheabsenceoftheinitialflaring,thehelicoidalmodelandthefirsthybridsolutionsufferfromthesamelowvalues i X of the magnetic field at 1 AU. However, by adjusting the parameters with a second hybrid solution, we are able to reproduce both the velocity and magnetic profiles observed by Ulysses and a reasonable description of the low corona, provided that a r a certain amount of energy deposit exists along theflow. Conclusions.Thepresentpapershowsthatanalyticalaxisymmetricsolutionscanbeconstructedtoreproducethesolarstructure and dynamics from 1 solar radius up to 1 AU. Key words.MHD - solar wind - sun - plasmas 1. Introduction Zouganelis et al., 2004) have been constructed to explore the kinetic aspects of the wind acceleration by supra- Since Parker’s model(1958),manystudies havebeenpre- thermal electrons in the collisionless region far from the sented to explain and predict the features and properties Sun. All models still have difficulties avoiding very high ofthe solarwind,mainlyfollowingtwodifferent,yetcom- temperature for the electrons. Other sources of heating plementary,approaches,kineticandfluidapproximations. such as turbulence (Landi & Pantellini, 2003) or Alfv´en Thesetechniquesareabletoreproducecertainaspects waves(Usmanov & Goldstein,2003;Grappin et al.,2002) of the observed solar wind but both show some limita- mayalsoexplaintheaccelerationbyloweringtheeffective tions,mainly dueto thecomplexityofthe severalacceler- polytropicindexoftheflow.Thispointisnotyetresolved ationmechanisms,the uncertaintiesconcerningthe origin andweshallnotaddressthisquestionhere.Insteadwewill of the fast solar wind, the associated problem of coro- invoketheneedforturbulenceorAlfv´enwavedampingin nal heating, etc. Different models have been presented our solutions. improving results of the acceleration. Two fluid models and more recently three-fluid models, (e.g. Ofman, 2000; Another approach consists of constructing MHD so- lutions to analyze the 3D structure of the wind, al- Send offprint requests to: Alexandre Aib´eo, e-mail: most independently of the heating source. Various mod- [email protected] els have been constructed, either 2-D ones able to de- 2 Aib´eo et al: Application of a MHD hybrid solar wind model toUlysses data scribe the low corona of the sun up to 10 solar radii data.The finalsolutionwasbasedonsolvingthe differen- (e.g. Pneuman & Kopp, 1971; Steinolfson et al., 1982; tial equations of the latter of these two models. Finding Cuperman et al., 1990 and references therein), the slow a solutionthatcomplies withthe constraintsgivenby the solar wind inside the brightness boundary in coro- data fit and the ones from its topological features is not nal streamers, (e.g. Nerney & Suess, 2005 and refer- easy. ences therein), 2-D ones for larger distances and mod- Regarding the limitations of LPT01 wind solution els for all range of distances. Those models were pro- (purely radial, yet very adaptable to the latitudinal de- posed because the flaring of the streamlines in polytropic pendences) and the ones of the similar study presented winds favors the acceleration. Recent observations by in SLIT05 (see also STT99; Sauty et al., 2002, hereafter Wang & Sheeley (2003) showed, however, that this may STT02 and Sauty et al., 2004,hereafter STT04), we take not be the case for the real solar wind. This favors a de- into account the advantages of both models by creating scription of the 3D structure of the solar wind using self- hybridsolutions.Theseusethe2.5DfeaturesoftheSTT04 similarMHDanalyticalsolutionsfornonpolytropicwinds model to describe the solarwind dynamics from the solar (Tsinganos & Sauty, 1992; Lima et al., 2001). In the first surface towardsthe Alfv´en sphere.Fromthe Alfv´enpoint of these two models it has been shown that the flaring of towards1 AU andbeyond, these hybridsolutions will use the lines may instead limit the acceleration of the wind. the advantagesof the LPT01model in fitting steepvaria- An increasing amount of observational data is now tions of velocity, density and magnetic field with latitude available. Ulysses measured for the first time the mag- and expressing the radial behaviour of the solar wind in netic field,the dynamicsandthe temperatureofthewind this region. However, we still solve the complete set of around 1 AU out of the ecliptic plane (McComas et al., MHD equations in the radial domain and not simply the 2000). Data from ACE (Stone et al., 1998), WIND Bernoulli equation along the streamlines. Thus, the solu- (Acuna et al., 1995), SoHO UVCS (e.g. Woo & Habbal, tion remains consistent everywhere. We discuss the prop- 2005), LASCO (e.g. Lewis & Simnett, 2002) and SoHO erties ofthe solutionsthus obtainedandphysicalgrounds CDS (Gallagher et al., 1999) are providing new insights for their limitations. into the origin of the solar wind within coronal holes. We maintain the criteria used to find a good solution Doppler Scintillation measurements (Woo & Gazis, 1994) fromthe Ulyssesdatafitandthe measuredvaluesofthe also brought new constraints to solar wind modelling. physicalquantitiesat1AU.Itwillbeshownlaterthat,for Semi-empirical models that use data to set bound- some sets of parameters, both the LPT01 model and the ary conditions for a numerical approach to the problem first hybrid model show the same problems mentioned in have also been proposed (e.g. Steinolfson et al. (1982); SLIT05,namelyinreproducingthevaluesofthemagnetic Sittler & Guhathakurta (1999); Groth et al. (1999)). field at 1 AU from Ulysses. These will be solved by a Neverthelesssome doubts onthe boundaries ofsome sim- judiciouschoiceofparametersinthesecondhybridmodel ulations are still present (Vlahakis et al., 2000). More that generates a solution consistent with Ulysses data. recently, some new developments have suggested that numerical simulations can benefit greatly from an an- alytical treatment (e.g. Keppens & Goedbloed, 2000; 2. Self-similar MHD outflow models from central Usmanov & Goldstein, 2003; Hayashi, 2005). Numerical rotating objects simulations are still quite time-consuming although this is rapidlyimproving.However,there areother limitations Thefollowingtwoaxisymmetricwindmodelsareobtained suchasmaintainingdivergence-freemagneticfields,limit- by self-consistently solving the full system of ideal MHD ingthe effectsofnumericalmagneticdiffusivityorsolving equations. In the present work we use spherical coordi- the 3D structure of the wind including rotation even at nates[r,θ,φ].Allquantitieshavebeennormalizedtotheir large distances. Note that the main problem with present values at the Alfv´en radius along the polar axis, similarly simulationsistheexistenceofanumericalmagneticdiffu- to SLIT05. They will be identified by the subscript *, i.e. sivity (e.g. Grappin et al., 2002). This is why we propose V∗, ρ∗ and B∗ for velocity, density and magnetic fields at to constructsemi-analyticalmodels whichareless sophis- the Alfv´en polar point, respectively. All equations will be ticated than numerical simulations but simpler to handle presented in a normalized form where the distance to the and more versatile. They also provide a complementary solar surface is related to the real distance by R r/r∗. ≡ approach. In the present work, that follows closely the work of 2.1. Model A with flaring streamlines Sauty et al. (2005 hereafter SLIT05) we focus on the dy- namics of the protons in the solar wind. We apply known InmodelAallthreecomponentsofthevelocityandmag- MHD analytical models to Ulysses data at solar mini- netic fields are accounted for (STT99, STT02, STT04). mum and test their advantages and limitations. We will Nevertheless, an expansion up to first order in latitude generate an exact wind solution based on the model of of the forces is performed by using harmonics with polar Lima et al. (2001 hereafter LPT01) and on the data fit values as references. Such a procedure makes the whole made in SLIT05. In SLIT05 two models (LPT01 and system analytical tractable and also describes the helio- Sauty et al. 1999 hereafter STT99) were used to fit the latitudinal variations of the wind quantities. The fields Aib´eo et al: Application of a MHD hybrid solar wind model toUlysses data 3 describing the outflow dynamics are Table 1. Parameters obtained by data fitting at 1 AU along the polar axis using the Ulysses hourly average fM2 cosθ Vr(R,θ) = V∗ R2 1+δfsin2θ (1) tdoartiaa,lepxlcaenpet.forBT,1AUwhichiscalculatedalongtheequa- M2 df sinθ p V (R,θ) = V (2) θ ∗ − 2R dR 1+δfsin2θ 1 fMp2/R2 Rsinθ Parameters Model B V (R,θ) = λV − (3) φ ∗ 1 M2 1+δfsin2θ (cid:18) − (cid:19) f B (R,θ) = B cosθ p (4) κf1AU 0.35 r ∗R2 δf1AU 1.95 B (R,θ) = B 1 df sinθ (5) f1AU 1.00 θ − ∗2RdR ǫ 5.64 1 f/R2 µ −0.18 B (R,θ) = λB − Rsinθ (6) φ ∗ 1 M2 (cid:18) − (cid:19) ρ ρ(R,θ) = ∗ 1+δfsin2θ (7) V1AU (km/s) 775 M2 n1AU (cm−3) 2.48 1 (cid:0) (cid:1) P(R,θ) = ρ V2 Π 1+κfsin2θ +C , (8) B1AU (µ G) 30.4 2 ∗ ∗ BT,1AU (µ G) 29.5 where V , V ,V are(cid:0)th(cid:0)e three compo(cid:1)nents(cid:1)of the velocity r θ φ field, B , B , B , the three components of the magnetic r θ φ field, ρ, the density, P, the pressure and C is a constant. There are three functions of R, namely M, f and Π. expresses the expansion factor. For a fully radial poloidal fieldline (i.e. an helicoidal pattern of the lines in 3D) we 2.2. Model B with helicoidal/radial streamlines have f = 1 which is the case of model B. The func- tion M(R) describes the poloidal Alfv´en Mach number Model B assumes a simpler geometry with radial stream which is unity at the Alfv´en radius. At this point the ki- andfieldlinesinthepoloidalplane(i.e.zeroθcomponents netic energy overtakes the magnetic one. A limitation of ofthe velocityandmagneticfields).Itis moreversatileat bothmodelscomesfromtheirself-similarnature.Thusthe reproducing steep latitudinal variations (LPT01). In this Alfv´enMachnumberisindependentoflatitudeandthere- case, the fields describing the outflow dynamics are fore the Alfv´eniso-surfaces are spherical. The functions Π,Π andΠ aredeterminedbynumericalandanalytical M2 1+µsin2ǫθ 0 1 V (R,θ) = V (9) techniques that are explained in STT02 and LPT01. r ∗ R2 s1+δsin2ǫθ 1 M2/R2 Rsinǫθ V (R,θ) = λV − (10) 3. A complete solution with helicoidal/radial φ ∗(cid:18) 1−M2 (cid:19) 1+δsin2ǫθ streamlines B B (R,θ) = ∗ 1+µsin2ǫθ p (11) In SLIT05 we fitted Ulysses data using the latitude de- r R2 q pendence of models A and B. We have shown that both 1 1/R2 B (R,θ) = λB − Rsinǫθ (12) modelsyieldsimilarparameters.ThesystemofODEswas φ ∗ 1 M2 (cid:18) − (cid:19) integrated exclusively using model A. Conversely, in this ρ ρ(R,θ) = ∗ 1+δsin2ǫθ (13) section we use the ODEs of model B to derive a full solu- M2 tion and compare the results with the ones from SLIT05. P(R,θ) = 1ρ V(cid:0)2 Π +Π s(cid:1)in2ǫθ , (14) The model flexibility provides a better fit of the latitu- 2 ∗ ∗ 0 1 dinal functions, which may be crucial in dealing with the where the same nota(cid:0)tionis used and(cid:1)Π and Π are func- poloidaldataat1AU.Yet,thissolutioncannotreproduce 0 1 tions of R. the flaring of the streamlines as they remain radial in the poloidal plane. 2.3. Geometry of the solutions 3.1. Method for a solution The relevant wind type solutions cross various critical points related to the non-linearity of the system of equa- The free parameters of the model (ǫ, δ and µ), the polar tions and its mixed elliptic/hyperbolic nature (see for in- valuesofthenumberdensity,radialvelocityandmagnetic stance Tsinganos et al., 1996; STT04). Each model is de- fieldat1AU,n ,V andB ,respectively,andthe 1AU 1AU 1AU scribed by three functions of R, M(R), Π(R) and f(R) for equatorial toroidal magnetic field at that same distance, modelA,M(R),Π (R)andΠ (R)formodelB.Thefunc- B , have already been constrained by the Ulysses 0 1 T,1AU tion f(R) characterizes the geometry of the fieldlines and data fitting procedures used in SLIT05. The end results 4 Aib´eo et al: Application of a MHD hybrid solar wind model toUlysses data are summarized in Table 1. From these constrained pa- Table 2. Input and output data for the critical wind so- rametersthelatitudefunctionsarewelldefined.Theradial lution calculated using model B, hybrid 1 and hybrid 2. dependences of the physical quantities are determined by integrating the ODEs of the model. From the knowledge of the values at 1 AU of the polar velocity, V , density, 1AU ρ , and magnetic field, B , it is possible to infer the 1AU 1AU input param. model B hybrid 1 hybrid 2 value of the Alfv´en number at that same distance, 4πρ V2 M2 = 1AU 1AU . (15) δ 1.95 2.90 0.49 1AU B12AU ǫ 5.64 5.64 5.64 µ −0.682 −0.406 −0.029 Simultaneously using Eq. (13) we obtain another impor- κ − 0.20 0.0123 tant reference value, 4πρ2 V2 ρ∗ = B1A2U 1AU . (16) At η=V1AU/V∗ 2.15 2.80 1.90 1AU 1 AU λ 0.1662 0.2468 0.1383 In the original paper (LPT01) the relations that rule ν 1.462 0.8872 0.3767 themodelarenormalizedtothesolarsurface.Thecritical V1AU (km/s) 775 775 775 solution for the solar wind is calculated based on three n1AU (cm−3) 2.48 2.48 2.48 B1AU (µG) 2.432 9.81 30.4 simple criteria. The first one is V (r/r =1)/V =1 and r ⊙ ⊙ BT,1AU (µG) 29.5 29.5 29.5 the other twoarethe continuityin the accelerationatthe Alfv´ensingularityandfastmagnetosonicseparatrix.Inthe present work we keep the two last criteria but Eqs. (9) to output param. (14) are normalized to the Alfv´en radius as in SLIT05. Thus, regarding the first criterion, the definition of the solar surface, r⊙/r∗ R⊙, poses a problem. In order to At η=V1AU/V∗ 2.151 2.193 1.908 ≡ calculate it we used the value atwhich the radialvelocity 1 AU R1AU 156.9 137.0 13.37 goes to zero or reaches its minimum value. As we expect the radial velocity to have a very steep variation at those V1AU (km/s) 775.4 607.6 779.4 distances,thecorrespondingerrorattheevaluationofthe n1AU (cm−3) 2.413 0.207 2.414 solar surface will be very small. The final solution should B1AU (µG) 2.369 0.641 29.8 also reproduce the measured values obtained by Ulysses TB1TA,U1A(U10(5µGK)) 239.0.4357 49..26446 4219..818 at 1 AU (mainly radial velocity, radial and toroidal mag- netic field and density). The total acceleration between the Alfv´enradius and 1 AU can be parameterized by, At the V∗ (km/s) 360.5 277.1 360.5 V Alfv´en n∗ (103 cm−3) 131.3 8.095 0.8423 η = 1AU . (17) radius B∗ (104 µG) 5.989 1.143 0.5435 V∗ BT∗ (103 µG) 9.953 1.360 0.278 Guessing this parameter, we obtain an initial value of V∗ T∗ (106 K) 9.892 3.902 3.597 and B =√4πρ V . From Eq. (11) it is possible to deter- ∗ ∗ ∗ mine the value of 1 AU in Alfv´enradius units, 1AU 215r B ⊙ ∗ R = = = . (18) 1AU r r B ∗(AU) ∗(r⊙) r 1AU The last equation gives the location of the solar surface since R = r /r = 215R . Already having the 1AU 1AU ∗ ⊙ anisotropy parameters, ǫ, δ and µ, we still need λ and ν. Combining Eqs. (11) and (12), assuming that we are at large distances (R >> 1) and since the lines are 1AU radial,whichisveryreasonableat1AU,weget,fromEq. (12) applied at 1 AU, on the equatorial plane, B M2 λ T,1AU 1AU . (19) ≃ B R Fig.1. Poloidal fieldlines and density contours of the so- ∗ 1AU lutionofTable 2formodelB.Distancesaregiveninsolar By definition, we also have radii. The solid circle line indicates the Alfv´ensingularity 2GM and the fast magnetosonic separatrix which are almost ν = . (20) sr∗V∗2 coincident. Aib´eo et al: Application of a MHD hybrid solar wind model toUlysses data 5 (a) (b) 103 361.4 361.2 radial velocity (km/s)110012 lap43etito50qulººuedaetor radial velocity (km/s)33663006..168 360.4 360.2 360 100 1 10 100 1.3681 1.3691 1.3701 1.3711 distance (solar radius) distance (solar radius) Fig.2. Panel (a) polar radial velocity as a function of distance. Panel (b) a zoom view of the singularities: the Alfv´enand the fast critical points, represented by and respectively. ∗ × Our procedure for finding a critical solution, using cal dashed line represents the Alfv´enradius, the accelera- model B, that fulfills all criteria is thus: tion at the critical points is clearly continuous. Note the presence of two different critical points, very close to one - ByfittingUlyssesdataat1AUobtaintheanisotropy another in Fig. 2 (b). However, searching for a converged parameters, ǫ, δ and µ, and the values B , B , 1AU T,1AU solution led us to this single set of parameters by using a ρ and V ; 1AU 1AU value of the radial magnetic field, B , of the order of 1AU - calculate R⊙, λ and ν with f1AU = 1 and using Eqs. one tenth of the value measured by Ulysses. A similar (18) to (20); discrepancy was also found in SLIT05 using the STT99 - with an initial guess of η, determine the values of V ∗ model instead of LPT01 model. Moreover,the parameter and B from Eqs. (15) and (16); ∗ that evaluates the anisotropy of the radialmagnetic field, - atthisstageitispossibletobuildacriticalsolutionus- µ, has also suffered a shift in its value (compare Table 1 ingthecriteriaofaccelerationcontinuityatthecritical to Table 2). As mentioned above we have tried to change points; the other input parameters and calculate their influence - this solution will give new values for the solar surface on the solution. The best option was to change those two radiusandforthedifferentphysicalquantitiesat1AU; parameters. Although these trials have been nearly ex- - iterate until the computed values of the solar surface haustive,degeneratedsetsoftheinputparametersforthe radiusandvelocityat1AUareclosetothefittedones. same critical solution may be possible. In Fig. 3 we show Two convergence criteria are inherent to this proce- the temperature profiles at various latitudes. The higher dure,thedistanceoftheAlfv´ensurfaceabovetheSun(or, effective temperature along the polar axis corresponds to equivalently, the value of the magnetic field strength at 1 the fast solar wind. At lower latitudes the lower temper- AU – see Eq. 18) and the velocity at 1 AU. Satisfying all ature is related to a mixing between the fast and slow the criteria only by changing η is not possible. Therefore, wind, which also corresponds to the lower velocities seen this can only be achieved by changing, in addition, at in Fig. 2 (a). The temperature distribution is similar to leastoneofthe parameters,thus releasingone ofthe con- the onepresentedinSLIT05althoughits maximumvalue straints. Considering that there are five constraints given is slightly better, around 10 106. × from the data fit, exploring all the parameter space is a We have also calculated the effective polytropic index formidabletask.Thesetofparametersconcerningthebest of this solution. After the temperature peak, it is almost possible solution is presented in Table 2. This will be dis- constant, between 1.1 and 1.3 (see Fig. 4). This value is cussed in the following section. quiteclosetothevalueinferredbyKoppandHolzer(1976) in their earlypolytropic model. Despite the difference be- tweenmodelAandBintheirgeometry,neithercanrepro- 3.2. Results ducefromtheobservationsthe highvalueofthemagnetic As can be seen in Table 2 we have constructed a solution field inferred at 1 AU. The calculations were made such wherethe convergencecriteriaarequite wellfulfilledwith that we keep the temperature as low as possible and a input/output ratios very close to unity for all physical reasonable value of the magnetic dipolar field at the base quantities.InFig.1weshowthefieldlinesandthedensity of the corona.We reproduce successfully the temperature contours in the poloidal plane. In Fig. 2, where the verti- and the velocity profile at 1 AU. It seems that the geom- 6 Aib´eo et al: Application of a MHD hybrid solar wind model toUlysses data Alfv´enradius, and the other from the Alfv´enradius to- wards infinity. In the first domain we use model A, since latitude 107 pole the 3D structure is more able to describe the wind struc- 45 º ture nearthe solarsurface.Althoughthe latitudinalfunc- 30º tionsofmodelAarenotveryversatileitdoesprovideafull equator K) 3D description of the flaring. In the domain further out, e ( ur wherethefieldlinesarealmostradialinthepoloidalplane, at mper106 wecanusemodelBandtakeadvantageofitsflexibilityin e te fittingverysteepvariationsofthephysicalquantitieswith ctiv latitude. The border between these two different domains e eff was arbitrarily set at the Alfv´ensurface. The major drawback of this construction is that we cannot guarantee continuity of all physical quantities ev- 105 erywhereexceptalongthepolaraxis.Generatingthecrit- ical solution with model A means that it must cross both 1 10 100 distance (solar radius) slow magnetosonic separatrix critical point and Alfv´enic singularpoint.Inaddition,thecriticalsolutionwithmodel B has to cross the Alfv´enpoint and a fast separatrix crit- Fig.3.Effectivetemperatureasafunctionofdistancefor ical point. Thus, a new feature of this hybrid model com- different values of latitude for the model B solution. The pared to our previous solutions (SLIT05 and Sect. 3) is vertical dashed line represents the Alfv´enradius. the crossing of the three usual MHD critical points. Such a situation was present only in the over-pressured solu- tionspresentedinSTT04.Sinceforthesolarwindthefast 1.4 point is close to the Alfv´enone, this was one more argu- menttoconstructthehybridsolutionstartingpreciselyat 1.2 this Alfv´enic transition. Moreover, in order to match the two solutions we must search for continuity of the physi- 1 cal quantities at the boundary as much as possible. This γdex, 0.8 means that at the Alfv´ensurface we ask for continuity of c in thedensity,pressure,velocityandmagneticfieldpluscon- pi ytro 0.6 tinuity of the accelerationand fieldline geometry. Strictly pol speaking, this can only be done along the polar axis be- 0.4 cause the latitudinal dependences of the physical quanti- ties are not identical in both models. Mathematically, we 0.2 have 0 ρm∗ odelA =ρm∗ odelB (21) 1 10 100 distance (solar radius) ΠmodelA+C =ΠmodelB (22) ∗ 0,∗ Fig.4.Polarpolytropicindexasafunctionofdistancefor V∗modelA =V∗modelB (23) the model B solution. The vertical dashed line represents BmodelA =BmodelB (24) the Alfv´en radius. ∗ ∗ dY modelA dY modelB = (25) etry does not control the decrease of the magnetic field dR|r∗ dR|r∗ but rather the temperature profile. It is even more sur- prising that in this LPT01 solution the density at 1 AU df modelA df modelB = =0 (26) remains at a reasonable level, thus both the velocity and dR|r∗ dR|r∗ the mass flux at 1 AU correspond to the observed val- ues. It is thus more consistent to build an hybridsolution The technique used to obtain a full hybrid solution is combining both models A and B. as follows. First, the radial velocity at the Alfv´enpoint, V , is determined by the same procedure as in the pre- ∗ vious section. Then, the model A critical solution has to 4. Hybrid solutions cross the slow separatrix critical point and the poloidal fieldlines have to be radial at the Alfv´enpoint, Eq. (26). 4.1. Method for a solution Thisyieldsthevalueofthe velocityslope(i.e.the acceler- For the construction of the hybrid model we consider ation) at that transition point. Similarly, the value of the two different domains, one from the solar surface to the accelerationat the Alfv´enpoint is determined by crossing Aib´eo et al: Application of a MHD hybrid solar wind model toUlysses data 7 reasonably good agreement with observations, namely at 1AU.ThepolytropicindexisshowninFig.8.Itshows,as expected, that a value around 1.2 is very well adapted to the solar wind, except in the low corona. However, as we shalldiscuss in nextsection,this firsthybridsolutionsuf- fersfromthe samedrawbacksasthepreviousone,despite its more sophisticated structure. In addition, the density is too low by one order of magnitude. Thus, although the temperatureprofileislowenoughtobeinagreementwith observations,the mass flux at1 AU remains too low.The low effective temperature does not prevent us from ob- taining large velocities but rather from obtaining large Fig.5.Plotofthefieldlinesconcerningthehybridsolution magnetic field at large distances. Thus we reconsidered anddensity contourspresentedin Table 2, column3,(hy- the values of some of the parameters, in particular the brid 1). Distances are given in solar radii. The black cir- value of the latitudinal dependence of the pressure which cles corresponding to the slow magnetosonic critical sep- isnotverywellconstrainedfromtheobservations,tocon- aratrix, the Alfv´ensingularity and the fast magnetosonic structanotherhybridsolutionmorefittedtotheobserved criticalseparatrix,whichalmostcoincidewitheachother. magnetic field. the fast separatrix critical point for the critical solution Anotherwayofanalyzingthedrawbackofthissolution of model B. In order for this slope of the velocity to be istoexaminetheconvergenceofthevaluer /r .Itrepre- ⊙ ∗ equal on both sides of the transition point, Eq. (25), the sentsby itselfthe convergenceofthe radialmagnetic field value of µ had to be changed from the one derived from intensity at 1 AU, Eq. (18).The convergenceof the values Ulyssesdata.Howeverthisparameterisnotverywellde- oftheAlfv´enMachnumberanddensityatthispoint,Eqs. termined and affects only model B. In model A its value (15) and (16), follows as a consequence. Ulysses data at is fixed to 1 by construction and cannot be fitted. For 1 AU lead to a very high value of R = r /r which, in − ⊙ ⊙ ∗ model A, knowing the slope of the velocity, Eq. (25), and turn, means that the acceleration of the wind up to the the geometry, Eq. (26), at the Alfv´enpoint, the value of Alfv´en speed should take place on a larger scale than the Π is fixed by crossing the Alfv´enpoint. Simultaneously, ∗ model predicts. Thus, a more satisfying solution should the value ofΠ is fixed by the conditionthat the pressure 0 display a lower total acceleration from the surface up to is zero at infinity. This determines the constant C by Eq. the Alfv´enpoint. The fully radial model used (see Sect. (22). Finally, the Alfv´endistance is fixed by the magnetic 3) is not able to produce that kind of behaviour. Some fieldstrengthat1AUforbothmodels,Eq.(18).Thus,the degreeofflaringis neededinordertoslowdownits accel- value of κ is determined such that the solution of model eration. Hence, the only way to deal with the problem is A matches the solar surface, R . ⊙ to decrease the acceleration zone by decreasing the value ofB at1AU,fromEq.(18).Ontheotherhand,inmodel r hybrid1,thehighdensitygradientneartheequator(acon- 4.2. Results for a hybrid solution - hybrid 1 sequenceofthe highvaluesofǫandδ)providesapoloidal Table 2 shows the input and output values for the most pressure towards the pole, improving the polar collima- important physical quantities regarding the first hybrid tion and subsequent acceleration of the wind. Therefore, solutionobtainedusingthetechniquepresentedinthepre- the equatorial gradient of the radial magnetic field must vious section. In this case convergence between the input increase(µ mustincrease),generatingahighermagnetic | | and output parameters is less satisfying than in Sect. 3. pressure towards the equator which counterbalances the This solution is hereafter referred to solution hybrid 1. previous effect. Figure 5 shows the geometry of the fieldlines for this so- lution. It clearly shows that beyond the Alfv´enpoint the fieldlinesbecome radialinthe poloidalplaneandthatthe Thiswindsolutionneedsavalueforthemagneticfield flaringzonenearthebaseofthewindiswelldefined.Thus anisotropy parameter, µ, very different from the one ex- conversely to SLIT05 where the dead zone was too ex- pected.Itcannoteasilydescribe thelatitudinalprofilesof tended,wehaveamorerealisticgeometry.Figure6shows thephysicalquantitiesfromUlyssesdataat1AU.Itsmain thepolarradialvelocitywheretheverticaldashedlinerep- limitation (besides the temperature, to which we will re- resentstheAlfv´enradius,whichcorrespondstotheborder ferlater)isitsdiscrepancyonthevalueofradialmagnetic between application of models A and B. field. Both radial and hybrid solutions fail completely in The presence of three critical points characterizes a reproducing the observed values of the the magnetic field different topology for this kind of wind solution (similar butthe hybridsolutionalsohasaprobleminreproducing cases were already discussed in STT04). Figure 7 shows the density at 1 AU. This lead us to construct the hybrid the profile of the temperature for this solution. It is in solution in a slightly different way. 8 Aib´eo et al: Application of a MHD hybrid solar wind model toUlysses data (a) (b) 3 10 280 279 m/s) 102 m/s) 278 k k y ( y ( cit cit 277 o o el el dial v 101 dial v 276 a a r r 275 274 0 273 10 1 10 100 1.555 1.56 1.565 1.57 1.575 1.58 distance (solar radius) distance (solar radius) Fig.6. Panel (a) the polar radial velocity as a function of distance for solution hybrid 1. The vertical dashed line is at the transition point (Alfv´enpoint). Panel (b), a zoom view of the three critical points is displayed. The slow magnetosonic, the Alfv´enand the fast magnetosonic critical points are labelled with +, and , respectively. ∗ × 107 1.4 1.2 K) 1 ective temperature (106 γpolytropic index, 00..68 eff 0.4 0.2 105 0 1 10 100 1 10 100 distance (solar radius) distance (solar radius) Fig.7.Effectivetemperatureasafunctionofdistancefor Fig.8. Radial profile of the polytropic index for solution the polar axis, for solution hybrid 1. The vertical dashed hybrid1.Theverticaldashedlinerepresentsthetransition line represents the transition point between the use of point between the use of model A and B. model A and B. Firstbecause,inthesuper-Alfv´enicregionwherethevalue 4.3. Results for a fully converged hybrid solution - of κ is determined for the Ulysses data, we use model B hybrid 2 in which we have no control over the latitudinal depen- denceofthepressure.Second,fittingthevalueofκinthis In Table 2 we show the input and output values of the domainisalmostirrelevantaswedonotreallycontrolthe most important physical quantities, for a second hybrid kinetic temperature (and pressure)which is the real tem- solution-hereafterreferredtosolutionhybrid 2.Although perature measured by the spacecraft. This discrepancy in we had to release some of the initial values of the param- theparametercaneasilybeevaluatedbycomparinginput eters as deduced from SLIT05, this new solution shows andoutputvaluesofthesameflowquantities.Figs.9and a much better agreement between the initial guesses and 10 show the geometryof the fieldlines andhow it changes the computed values. from model A to model B. A new feature of this solution For this solution, we had to change the value of κ and can be seen in Fig. 11 (a) - a zone where the radial ve- of δ, although in a less dramatic way.Changing the value locity attains a local minimum, close to the Alfv´enpoint. of κ in the sub-Alfv´enicregime is not a serious problem. Figure 12 displays the effective temperature profile along Aib´eo et al: Application of a MHD hybrid solar wind model toUlysses data 9 the polar axis. This plot clearly shows that the kinetic pressure alone cannot account for the effective tempera- ture.Figure13showsthe correspondingpolytropicindex. The excess between this temperature and the observed one can be accounted for, as in SLIT05, assuming extra pressure from Alfv´enwaves or turbulent or ram pressure (Fig. 14). We have calculated the amplitude of the ram velocity/magnetic field fluctuations, if the effective tem- peraturewehavecalculatedisassumedtobethe resultof turbulence/Alfv´enwaves. For ram pressure we assume: 1 Pram = ρδV2, (27) Fig.9.Fieldlinesanddensitycontoursconcerningthehy- 2 brid solution presented in Table 2 (hybrid 2). Distances and for the Alfv´enpressure we take are given in solar radii. The three black circles represent δB2 the three surfaces: the slow magnetosonic separatrix and PAlfven = , (28) Alfv´ensingularity,whicharealmostcoincident,andafast 8π magnetosonic separatrix. These calculations are arbitrary and a mixing of vari- ous processes is probably the source of the extra pressure that accelerate the fast wind. However it gives an order of magnitude of the fluctuations needed. Comparing the variousplotsa),b),c)andd)inFig.14,weconcludethat Alfv´enwaves are more appropriate to explain the acceler- ation in the sub-Alfv´enic part where the magnetic field is dominant. Conversely, turbulence and fluctuations of the velocity may account for the acceleration in the super- Alfv´enicregion.We arriveat this conclusiononly because thecalculatedfluctuationsofthemagneticfieldinthesub- Alfv´enic region are smaller than the calculated turbulent velocity field and the reverse holds in the super-Alfv´enic part. This is the best way to minimize the amplitude of Fig.10.Fieldlinesanddensitycontoursforsolutionhybrid 2, the fluctuations in both regions. A mixture of the two close to thesolar surface. Distances are given in solar radii. components is probably more realistic but this needs a more detailed model to interpret the role of turbulence in heating the flow. physics controllingthe hybridmodel forcedus to adaptit This new hybrid solution generates a field geometry in order to obtain the requiredfeature. Reminding that κ that is continuous at the transition point (it is still not andδ characterizetheanisotropyofthepressureandden- differentiable and kinks in the field are unavoidable) and sity, decreasing both parameters will lead to a decrease showsfeaturesexpectedforthesolarwind(Figs.5and9). of the pressure gradient towards the pole, which enables It is also capable of reproducing almost all Ulysses data the wind to accelerate more slowly (from the solar sur- at1AU.Despitetheslightdifferencebetweenthevaluesof face to the Alfv´enpoint). As a consequence of accelera- the anisotropy parameters when calculated by fitting the tion continuity atthe transitionpoint, µ also diminishes | | data(Table1)andthe onefromthe criticalsolutionitself whichmeansthatthe magneticpressuregradienttowards (Table2),mostofthelimitationsoftheprevioussolutions the equator, outside the Alfv´ensphere (in the fully radial have been solved. Such a discrepancy should not be very zone of the model), also decreases. For the dynamics of important since the new values can easily be fitted (with the radial part of the hybrid model, the wind velocity is some degree of accordance) to the observed data (see for expected to be higher in order to satisfy the values at 1 instance the fit of the density in Fig. 15 and of the radial AU and so it needs to accelerate the wind. This leads to velocity in Fig. 16). The ratio between the input and the a decrease in the magnetic pressure towards the equator output values for the most important physical quantities and thus a decrease of µ. is very close to unity and all the continuity criteria are Of course, there is a price to pay to fit all data at 1 satisfied. Nevertheless, κ, δ and µ have values departing AU. Thus, this hybrid model has two major drawbacks. fromthe expectedones.Thevalue ofµisthe resultofthe Thephysicalquantitiesarediscontinuousatthetransition transition conditions stated in Eqs. (21) to (26) and has radius except for the polar ones. This is a natural conse- been calculated accordingly. quenceoftheanalyticalexpressionsthatdescribetheflow, Thevaluesofthemagneticfieldintensityat1AUcon- Eqs.(1) to (14) and the temperature behaviour.The first strainthevalueofR ,andthereforethelengthofthewind one is solved only for values of ǫ = 1. The second and ⊙ acceleration zone (or the dead zone). Consequently, the more serious drawbackis the very high effective tempera- 10 Aib´eo et al: Application of a MHD hybrid solar wind model toUlysses data (a) (b) 411 3 10 410.5 m/s) 2 m/s) 410 ocity (k 10 ocity (k 409.5 el el al v al v adi 101 adi 409 r r 408.5 0 10 408 1 10 100 15.5 16 16.5 17 distance (solar radius) distance (solar radius) Fig.11. Panel (a) the polar radial velocity as a function of distance for solution hybrid 2. The vertical dashed line is at the transition point (Alfv´enpoint). Panel (b), a zoom view of the three critical points: the slow magnetosonic, the Alfv´enand the fast magnetosonic critical points, labelled with +, and , respectively. ∗ × 107 1.5 1 K) effective temperature ( γpolytropic index, 0.5 0 106 −0.5 10 100 10 100 distance (solar radius) distance (solar radius) Fig.12. Effective temperature as a function of distance Fig.13.Radialprofileofthepolytropicindexforsolution for the polar axis, for solution hybrid 2. The vertical hybrid2.Theverticaldashedlinerepresentsthetransition dashed line represents the transition point between the point between the use of model A and B. use of model A and B. 5. Conclusions ture. This can be explained only if we calculate the heat flux using a reasonable kinetic theory (Zouganelis et al., From the constrained parameters obtained after fitting 2005) together with solving a full energy equation. This Ulysses data (SLIT05) we were able to build different amounts to invoking a non thermal heating term, a dif- criticalsolutionsforthesolarwind.Thefirstwasobtained ficult task that we postpone for future work. In Fig. 12, using a purely radial field (model B). The remaining two we see how the energy equation can be essential. The ab- solutions where constructed as hybrid ones incorporat- sence of the an abrupt increase of the temperature very ing an inner region where model A (with flaring stream- close to the surface in other models, such as the one pre- lines) was used and an outer one with model B. Thus, sented in SLIT05 and the one presented in Sect. 3 of the we combine the advantages of model B of reproducing present work might be explained by solar surface being highly adaptablefunctions oflatitude andthe advantages much closer to the Alfv´enradius and therefore the prob- of model A of ensuring adequate flaring of the fieldlines lems had not emerged yet. Nevertheless, high tempera- to get a more realistic geometry of the overallsolar wind. tures are reached (for an overall behaviour of the wind Thesetwodistinctmodelswerecoupledusingwelldefined solution) as a consequence of high values of the magnetic domains for each one and a suitable transition zone, the field at 1 AU and not necessarily high values of velocity Alfv´enradius. Both model B (used by itself) and the hy- as one may expect intuitively. brid model (A and B coupled) were used to generate a