Coupling the Solar Dynamo and the Corona: Wind Properties, Mass, and Momentum Losses During an Activity Cycle Rui F

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Coupling the Solar Dynamo and the Corona: Wind Properties, Mass, And Momentum Losses During An Activity Cycle Rui F. Pinto, Allan Brun, Laurène Jouve, Roland Grappin

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Rui F. Pinto, Allan Brun, Laurène Jouve, Roland Grappin. Coupling the Solar Dynamo and the Corona: Wind Properties, Mass, And Momentum Losses During An Activity Cycle. The Astrophysical Journal, American Astronomical Society, 2011, 737 (2), pp.72. 10.1088/0004-637X/737/2/72. cea- 00828283

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. arXiv:1106.0882v1 [astro-ph.SR] 5 Jun 2011 tde aei ml atsa oan hc in- which domains cartesian small in made studies 2007). 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Rui ≪ and 1 β oadGrappin Roland Laur oapa nApJ in appear To ≫ ABSTRACT 1) eo,LbrtieUiese here,Osraor eM de Théories, Observatoire et Univers Laboratoire derot, n Jouve ene ` DdrtCR/IS,911GfsrYet,France Gif-sur-Yvette, 91191 INSU, CNRS/ -Diderot ispto.Teidcieato ftecmlxfluid complex ohmic the against of them action sustain inductive and The components field three acts the dissipation. magnetic process regenerate the dynamo to a interior of that solar is the field in magnetic scale large a field. playing toroidal poloidal tachocline organising the and in with toroidal role separated, central the are of field generation that global assumes of model locations This the 1993). par- received (Parker In has attention paradigm sun. much dynamo the interface the of solar at activity the is magnetic ticular, dynamo intense “fluid” the multi- This of on dynamo. origin rely solar origin of models and D evolution structure, its dynamo netism, hand, observables photospheric other the reproduce detail. greater to On in 2006; able in Duvall 2003). are field & 2000, models magnetic Kosovichev al. the 2006; et of Antia (Kosovichev properties sun to the though, the of used, some been deduce have measurement. techniques and/or Helioseismology observation direct by a unreachable at properties wind’s slowly solar the scale. and capture global field to few though, magnetic unable, varying a 2009; are 2009). studies al. al. of Wedemeyer-Böhm among by et These et review Leenaarts order 2005, also 2009; see al. of Vöglerothers; al. et typ- 2008; et being the (Rempel al. Mart´ınez-Sykora et Mm size domain, of the domain tens in numerical layers ical photospheric the clude h lsia xlnto o h ylcatvt fthe of activity cyclic the for explanation classical The mag- solar inner the of understanding our Currently is sun the inside field magnetic the of structure The mnu u n antcbraking magnetic and flux omentum o sdtrie ytelatitudinal the by determined is ion . aasa,France Palaiseau, t eeslhpesa different at happens reversal ity r fteAfńsrae Wind Alfvén surface. the of try re nlttd n ntm in time in and latitude in aries eeouino h oa wind solar the of evolution he i ieai yaocode dynamo kinematic ric DP.Tecomputations The (DIP). nce CYCLE Y uo,5Place 5 eudon, on- 2 Pinto et al. motions would thus be responsible for the vigorous re- of the solar cycle. The amplitude and latitudinal dis- generation of magnetic fields and for its nonlinear evo- tribution of the solar wind velocity and mass flux de- lution in the solar interior (see Charbonneau 2010 and pend on parameters such as the positions of the wind Miesch 2005 for recent reviews on the subject). Un- sources at the surface, the local magnetic field strength derstanding how these complex physical processes op- and the expansion factor of each particular coronal flux- erating in the solar turbulent plasma non-linearly in- tube the wind flows along. The details about the coro- teract is very challenging. One successful and power- nal heating mechanisms, most notably the amplitude ful approach is to rely on multi-dimensional magnetohy- and location of energy dissipation, are also of impor- drodynamic (MHD) simulations. In this context, two tance. Leer & Holzer (1980) pointed out that heating types of numerical experiments have been performed below or above the critical sonic point may produce dif- since the 70’s: kinematic mean-field axisymmetric dy- ferent effects on the wind velocity, and that high speed namo models which solve only the mean induction equa- winds require energy deposition in the supersonic region. tion (Steenbeck & Krause 1969; Roberts 1972; Stix 1976; Hansteen & Leer (1995) showed that the mass flux does Krause & Radler 1980) and full 3D global models which not depend strongly on the mode and location of en- explicitly solve the full set of MHD equations (Gilman ergy deposition, but rather on the amplitude of the en- 1983; Glatzmaier 1985; Cattaneo 1999; Brun et al. 2004). ergy flux. Note also that the heat flux profile could be Clearly, both approaches are complementary and are related to the magnetic field’s amplitude and geometry needed to better understand the magnetic solar activ- (Cranmer et al. 2007; Pinto et al. 2009). The resulting ity. Recent progress have been made with 3D numeri- angular momentum losses and wind’s braking torque (ap- cal models of magnetic stars. Large-scale magnetic cy- plied on the sun) depend on wind’s velocity and mass cles are starting to be found in simulations of the Sun, flux together with the (time varying) geometry of the such as those performed by Ghizaru et al. (2010), or of Alfvén surface (the surface at which the wind’s velocity solar-like stars (Brown et al. 2011). However, butterfly equals the Alfvén velocity; see definitions in 5). Some diagrams similar to the solar observations — with the authors have studied the influence of the topology§ of poloidal field field reversal happening when the toroidal coronal field in the properties of stellar winds and re- field is at its maximum — are still difficult to repro- sulting braking torques, but mostly by using simplified duce. Kinematic mean-field models and their associated configurations (e.g, dipolar versus quadrupolar field, as simplifying assumptions have thus been used extensively in Matt & Pudritz 2008). Others have furthermore con- to reproduce several features of the large-scale solar cy- sidered how the presence of strong magnetic spots (lo- cle. In particular the use of a differential rotation profile calised magnetic flux enhancements) specifically affects inferred from helioseismology associated with an alpha- the angular momentum loss rate (e.g, Aibéo et al. 2007; effect (due to the helical turbulence of the stellar convec- Cohen et al. 2009). tive envelope) antisymmetric with respect to the equator More detailed studies of the global solar magnetic enabled Charbonneau & MacGregor (1997) to produce structure usually adopt surface magnetogram data for a solar-like butterfly diagram with the ingredients of the the radial component of the field as a lower bound- modern interface dynamo. A similar model will be con- ary. The atmospheric magnetic field’s geometry is de- sidered in this work to catch the large-scale behaviour of duced using potential field extrapolation techniques. the inner solar magnetic field. The coronal and heliospheric hydromagnetic condi- Above the photosphere, a very complex structure tions are then deduced either by using a set of semi- comprising open magnetic flux-tubes (coronal holes) empirical relations (Arge & Pizzo 2000; Luhmann et al. and magnetic loops with different length-scales arises 2002; Wang & Sheeley 1992; Schrijver & DeRosa 2003), (magnetic carpet, canopy, coronal loops) (Stix 2002; or by finding stable MHD solutions for the mapped fields Aschwanden 2005, and references therein). The wealth (e.g, Usmanov 1993; Hu et al. 2008). The usage of mag- of ground and space based observational data gathered netogram data as lower boundary conditions encounters in the latest years has been bringing up a great in- a few difficulties. Namely, only the line-of-sight com- sight on the magnetic structure in the lower atmospheric ponent of the field is available, making the estimation layers (chromosphere and lower corona). The proper- of the radial components difficult at the polar regions. ties of the magnetic field in the (optically thin) higher The standard PFSS technique (standing for “potential corona are harder to determine observationally, though. field source surface”) further assumes the surface field Space-borne in-situ measurements may introduce further is strictly radial. One still needs to assume an outer constraints about how the solar wind properties con- boundary condition which imposes the magnetic field to nect to the coronal structures. Remarkable examples of be purely radial as well at a source-surface (typically, a these were given by the Ulysses consecutive polar orbits spherical surface placed at about r =2.5 R⊙). This con- (McComas et al. 2003; Issautier et al. 2004, 2008). The dition emulates the field-line opening usually caused by SoHO spacecraft further provided complementary in-situ the solar wind flow, while keeping the global field at its and solar surface (and low coronal) observations. Future lowest (current-free) energy state. These studies essen- missions such as the Solar Orbiter should provide more tially provide quasi-stationary predictions of the coronal refined data to complete the scenario. These should com- fields at a given moment. Despite all the simplifications bining high resolution imagery and spectroscopic data, and assumptions this family of models has been success- magnetogram and in-situ measurements of the wind’s ful in predicting qualitatively the coronal magnetic field’s properties in an orbit with very close helio-synchronous topology (observed, for example, in white light during passes and varying orbital inclinations. solar eclipses) and became a standard to which other The solar wind’s outflow properties depend on the par- types of model can be compared. Their MHD counter- ticular geometry of the coronal field at each moment parts are limited in different ways. Beyond the much Coupling the solar dynamo and the corona 3 increased computational power they require, they rely the external component of the coronal magnetic field. on phenomenological assumptions for the heat transport Further details about both numerical codes are given in the corona (as the heating processes are still under hereafter, in 2.1 and 2.2. Details about the coupling of debate). These models show, nevertheless, some quan- the two codes§ are given§ in 2.3. titative differences when compared to the PFSS ones. § The relative sizes of the streamers can be different (usually more elongated) and the heliospheric magnetic field 10 tends to be more uniform (as it is known to be). See, for 3.00 example, Riley et al. (2006) for a comparison between these two families of models. A third type of models — 2.00 nonlinear force-free field (NLFFF) models — places it- 5 self somehow in between the two previously described, 1.00 in the sense that it tries to add some complexity to the 1 1 0 0.00 PFSS scenario while avoiding the issues found by global -1 MHD computations. These require, though, good qual- -1 ity vector magnetogram data at the surface of the sun -1.00 and still find some difficulty in accurately predicting real -5 solar features (DeRosa et al. 2009). -2.00 In this paper, we investigate the influence of the cyclic evolution of the large-scale magnetic field produced by -3.00 -10 the solar dynamo on the solar wind properties. We chose 0 2 4 6 8 10 not to perform this study based on surface magnetogram Figure 1. Representation of the numerical domains of STELEM magnetic field data (as in Wang et al. 2003; Hu et al. (left) and DIP (right) at the same instant. The colour scale in 2008; Luhmann et al. 2009, for example), but rather pur- the left figure shows the toroidal magnetic field Bφ in the convec- sue a different path: our source fields are the result tion zone. Black lines are (poloidal) magnetic field lines (dotted and continuous lines indicates CW and CCW field-lines, respec- of well-tested and validated kinematic dynamo models tively). The white lines in the right figure are magnetic field lines, (Jouve et al. 2008). An isothermal MHD model of the while the colour-scale represents both the wind’s velocity in units solar wind and corona is used to produce a temporal se- of Mach number and the open field’s polarity (blue is negative, red quence of steady-states spanning a complete activity cy- is positive). cle. The focus here is on estimating the wind’s velocity, mass and angular momentum flux spatial profiles as they vary during an activity cycle in response to the variations 2.1. STELEM: a representative dynamo solution of the magnetic field’s topology, rather than reproducing a particular solar cycle. The variability of global proper- To investigate the global solar cycle features produced ties such as the sun’s mass loss rate and wind’s breaking by dynamo models, we start from the hydromagnetic in- torque are studied in regards to the activity cycle. Also, duction equation, governing the evolution of the mag- we expect to gain a deeper insight on the connections netic field B in response to advection by a flow field U between the sub-surface and coronal physical processes and resistive dissipation. by proceeding this way. ∂B The remainder of this manuscript is organised as fol- = (U B) (η B) lows: the methods and numerical codes used are pre- ∂t ∇× × −∇× ∇× sented in 2. The results are described thoroughly in 3 § § As we are working in the framework of mean-field the- (coronal magnetic field evolution), 4 (solar wind speed) ory, we express both magnetic and velocity fields as a sum and 5 (mass and angular momentum§ flux, breaking § of large-scale (that will correspond to the mean field) and torque). A discussion follows in 6 and a brief summary small-scale (associated with fluid turbulence) contribu- in 7. § § tions. Averaging over some suitably chosen intermediate 2. COUPLING SOLAR DYNAMO AND WIND MODELS scale makes it possible to write two distinct induction equations for the mean and the fluctuating parts of the We used two 2.5D axisymmetric MHD codes. The first magnetic field. one — STELEM (Jouve & Brun 2007, and 2.1) — com- § A closure relation is then used to express the mean elec- putes mean-field kinematic MHD dynamo solutions given tromotive force in terms of the mean magnetic field, lead- the meridional circulation and differential rotation pro- ing to a simplified mean-field equation (Moffatt 1978). files in the solar convection zone. A Babcock-Leighton The prescribed mean velocity field will here only con- or an α source terms generate the poloidal magnetic sist in its longitudinal component (the solar differential field. The second code — DIP (Grappin et al. 2000, and rotation) and the magnetic diffusivity is assumed to be 2.2) — computes the temporal evolution of an MHD § constant. The source term for the poloidal field is linked solar corona with a self-consistent wind. In the latter, to the turbulent helical motions within the convection the magnetic field separates into two components: an 0 zone. We thus obtain a simple αΩ dynamo model with imposed stationary external potential field B (whose constant diffusivity. sources lie within the sun), and an induced component Working in spherical coordinates and under the as- produced by the flows within the numerical domain. The sumption of axisymmetry, we write the total mean mag- dynamo magnetic fields produced by STELEM match a netic field B as potential field at the surface of the sun. This potential source field can then be directly transmitted to DIP as B(r,θ,t)= (A (r,θ,t)eˆ )+ B (r,θ,t)eˆ ∇× φ φ φ φ 4 Pinto et al.

Reintroducing this poloidal/toroidal decomposition of value of Ωeq = 460 nHz, in agreement with observations). the field in the mean-field induction equation, we get The spatial resolution is 128 128. two coupled partial differential equations, one involving × the poloidal potential Aφ and the other concerning the toroidal field Bφ ∂A 1 φ = C αB + ( 2 )A (1) ∂t α φ ∇ − ̟2 φ

∂B φ = C ̟( (̟A ê )) Ω ∂t Ω ∇× φ φ · ∇ 1 + ( 2 )B (2) ∇ − ̟2 φ where ̟ = r sin θ, Ω is the differential rotation and α is the α-effect. All these quantities are now dimensionless, thanks to the presence of the two Reynolds numbers CΩ = 2 ΩeqR⊙/ηt, and Cα = α0R⊙/ηt, ηt being the turbulent Figure 2. Time-latitude diagram of the surface field. The upper magnetic diffusivity, Ωeq a measure of the rotation rate panel shows Bθ while the lower panel shows Br. Red (blue) colours at the equator and α0 a measure of the intensity of the indicate positive (negative) values of the field. The white dashed alpha-effect. The product of these two numbers gives us lines span the time interval which was chosen to be incorporated the dynamo number which measures the efficiency of the in the DIP code to compute the solar wind evolution and mark the source terms to make the magnetic energy grow in time instants shown in the following figures. against Ohmic diffusion. Equations (1) and (2) are solved in an annular merid- Figure 2 shows the temporal evolution of both the lati- ional cut with the colatitude θ [0, π] and the radius tudinal and the radial field at the surface, at all latitudes. We have here the evidence of a cyclic magnetic field: the r [0.65, 1] R⊙ i.e from slightly∈ below the tachocline surface poloidal field changes sign regularly and at dif- (e.g.∈ r = 0.7 R⊙) up to the solar surface (see Figure 1, left panel). At θ = 0 and θ = π boundaries, both ferent times depending on the latitude of interest. This simple model does not aim at reproducing the solar dy- Aφ and Bφ are set to 0. At r = 0.65 R⊙, we compute a perfect conductor condition. At the upper boundary, namo in its finest details. The goal here is rather to we smoothly match our solution to an external potential assess the influence of a cyclic poloidal background field (having the same general properties as that in the sun) field, i.e. we have vacuum for r R⊙. The STELEM (STellar ELEMents)≥ code uses a finite on the wind’s properties. The dashed white lines labelled element method in space and a third order scheme in t1 and t6 in the figure indicate the beginning and the end time (Burnett 1987, Jouve & Brun 2007). of the particular time interval used to compute the so- The principle of the finite element method is to look for lar wind evolution. This time interval spans an entire solutions of the weak formulation of the equations, here cycle period and is roughly located between two activity equations (1) and (2). Those solutions are taken to be minima. linear combinations of well-chosen trial functions. In our 2.2. DIP: wind model case, they are Lagrange polynomials of degree 1 and the elements are rectangles in the ( cos θ, r) plane. Apply- The DIP code is a 2.5 D axisymmetric model of the ing this spatial method to our equations− results in a sys- solar corona obeying the compressible MHD equations tem of first order ODEs in time governing the evolution for a one-fluid, isothermal and fully ionised plasma. The of the coefficients of the linear combinations. The tem- continuity and momentum equations are poral scheme that we use is adapted from Spalart et al. ∂ ρ + ρu = 0 (3) (1991). It is similar to a Runge-Kutta 3 method and is t ∇ · ∂ u + (u ) u = thus explicit and of order 3. t · ∇ The STELEM code is here used to produce a cyclic P J B ∇ + × g + ν 2u . dynamo field within the model convection zone, whose − ρ µ0ρ − ∇ values at the top of our domain will be reintroduced in the DIP code described below. We chose a simple αΩ We set γ = 1 and a uniform temperature T = 1.3 MK dynamo model to produce our representative solution, in all the corona. The gas pressure is deduced from the with physical source terms consistent with observations. equation of state The rotation profile is deduced from helioseismic inver- 2 sions and the α-effect is antisymmetric with respect to P = ρkB T, the equator and positive in the Northern hemisphere (in mH agreement with the preferred handedness of the turbu- valid for a fully ionised hydrogen plasma. The isother- lent helical convective motions). This is similar to case mal approximation discards a complete treatment of A of (Jouve et al. 2008). The following values were used the energy fluxes in the corona. But we should note for the various parameters: Cα = 0.385 (critical value firstly that the corona is nearly isothermal (at least for which dynamo action just starts to compete against in the first 12 R⊙). Secondly, assuming γ = 1 is a Ohmic diffusion), C = 1.4 105 (corresponding to a proxy to the actual thermal state of the coronal plasma, Ω × Coupling the solar dynamo and the corona 5 summing-up the combined effects of the thermal con- i.e, mass can flow through them and waves are not duction and the (still debated) heating source. The spuriously reflected there (see Figure 1, right panel). choice of a particular value for the coronal tempera- This is achieved by writing the MHD equations in ture is somewhat arbitrary, though. The temperature their characteristic form. In simple terms, this con- T = 1.3 MK chosen here is justified both by represent- sists in projecting the system of equations in terms of ing an average coronal value in the first 10 R⊙ (see, the primitive quantities ρ, U, B into an equivalent ∼ { } e.g, results by Hansteen & Leer 1995; Endeve et al. 2003; system defined in terms of characteristic variables Li Guhathakurta et al. 2006; Pinto et al. 2009, among oth- (Thompson 1987; Vanajakshi et al. 1989; Poinsot & Lele ers) and empirically as it produces correct wind solu- 1992; Roe & Balsara 1996). The evolution equation for tions. As discussed later in the text, the wind’s mass flux these variables explicitly define the time-evolution of the depends both on the coronal temperature and on wind MHD system in terms of wave-modes propagating up- geometrical expansion factors. We focus here on the geo- wards and downwards, or equivalently, incoming and out- metrical effects linked to the activity cycle and keep our going modes. We must then constraint all the incoming reference temperature fixed throughout the simulations. modes, and let the others free. Plasma is free to flow The magnetic field B decomposes into a potential ex- through and its properties to vary in time, the actual ternal component B0 and on a component b induced boundary values at each moment depending on the cur- by the flows. The former only has poloidal compo- rent state of the system. The mass fluxes and velocities nents (r, θ), while the latter comprises both poloidal and (or their gradients) are, therefore, not arbitrarily set at toroidal components. The total magnetic field is then the numerical boundaries. This is a critical feature: the defined as actual solar wind velocities and mass flux at each point 0 B = B + bpol + bφ . (4) of the surface will automatically match each unique transonic and transalfvénic wind solution. There is no need b Φ Furthermore, pol = . We integrate the evolution to set in advance arbitrary values for the density, ve- ∇× Φ equation for the magnetic potential locity or mass flux at the numerical boundaries. This 2 ∂tΦ = urBθ Bruθ + η Φ . (5) means no spurious boundary layers will form there and − ∇ that the resulting mass fluxes will not be artificially con- The poloidal components of b are then computed as strained. We note that the resulting surface density and 1 mass flux do not vary by large factors (not more than b = (Φ cot θ + ∂ Φ) , r r θ 10% in all latitudinal domain, and less than 6% within open∼ field regions at all moments of the cycle). Further- Φ bθ = ∂rΦ . more, this type of boundary conditions better describes − r − dynamic finite-frequency phenomena than any type of The azimuthal component of the induced field comes di- non-transparent (i.e, “rigid”) conditions (Grappin et al. rectly from the induction equation (in the φ direction 2008). only), that is The solar wind develops into a stable transonic and transalfénic solution in the open field regions after a pres- ∂ b = (u B)+ η 2B . t φ ∇× × ∇ φ sure (sonic) perturbation is applied at the outer boundary. This perturbation propagates inwards until both This method guarantees that the solenoidal condition ∇· the sonic and alfvénic surfaces appear (and are enclosed B = 0 is satisfied at all times. This papers focus only on in) within the domain. From then on, the solar wind the poloidal components of the coronal field, though (as profiles evolve towards a unique stable state (Velli 1994; explained hereafter, in 2.3). § Grappin et al. 1997). Note that the frontiers between The diffusive terms are adapted so that grid scale (∆l) open (coronal holes) and closed flux regions (streamers) fluctuations are correctly damped. The kinematic vis- 2 are not set in advance, but they are rather a result of cosity is defined as ν = ν0 (∆l/∆l0) , typically with the competition between the magnetic tension and the 14 2 −1 2 ν0 = 2 10 cm s and 0.01 . (∆l/∆l0) . 10. dynamical pressure in the wind flow. In other words, the The simulations× presented· in this paper were performed solar wind settles where the magnetic field is incapable with a vanishing η, so as to approach as much as pos- of containing the outward flow and forces the magnetic sible the limit of ideal non-resistive MHD. Additional field to align with the radial direction. diffusive terms are used. These are implicit numerical Other aspects of the numerical model are more thor- filters (applied over u, b and Φ) which dissipate mostly oughly discussed in Grappin et al. (2000). at the grid scale and minimise the dissipation of large 2.3. scales fluctuations (Lele 1992). This filtering scheme al- Coupling method 0 0 lows the diffusive parameters ν and η to be lowered while We used a time-series Br (t) ,Bθ (t) derived from a avoiding spurious Gibbs fluctuations, and allow for lower STELEM run (Jouve & Brun{ 2007) which} correctly pro- mid-scale damping (Grappin & Léorat 2001). Note that duces a cyclic behaviour of the poloidal magnetic field. actual kinetic dissipation should happen at scales much We used a simple αΩ dynamo model with source terms smaller than the grid size, anyway. for both the toroidal field and the potential Aφ. We used a 5122 grid which is uniform in latitude and The time-series was scaled in order to span an 11-year non-uniform in radius. The grid’s cells radial extent is period, sampled with a time step of 6 months. The verti- −3 δr = 6.5 10 R⊙ at the lower boundary and δr = cal dashed white lines labelled t1 and t6 in Figure 2 show −1 1.0 10 × R⊙ at the upper boundary. the beginning and end of the time-series (t = 0 and × 1 Both the upper and the lower numerical boundaries t6 = 11 yr). These are the moments when the poloidal (respectively at r = 15 and 1.01 R⊙) are transparent, field is at its simplest configuration (i.e, lowest multipo- 6 Pinto et al. lar order). For simplicity, we will call these moments the “giant polar plumes” in Pinto et al. 2010 and the “activity minima”, while the magnetic field’s polarity in- “pseudo-streamers” in Wang et al. 2007). Loop struc- version phase (between lines t2 and t5 in the same figure) tures placed inside stagnant zones (i.e with no wind flow, will be called “activity maximum” hereafter. The time- as inside large streamers) mostly maintain their potential series comprises 22 equally spaced samples, but we will field configuration. refer to a subset of 6 samples (t1 to t6) for illustrative We start the computation at the moment when the purposes throughout the text. We let the solar wind fully magnetic structure of the corona is at its simplest (Fig- develop in the domain and reach a steady state (see 2.2 ure 3, first panel, t1; also first white line in Figure 2). for further details on the formation of the wind.) The§ One and only large equatorial streamer extends from the amplitude of the external magnetic field B0 was scaled surface up to nearly 5 R⊙, where the heliospheric cur- so that the total field would match coronal amplitudes rent sheet starts. The frontiers between the streamer (the scaling factor being kept constant during the whole and the coronal holes cross the surface at latitudes 60◦ cycle). Then, for each consecutive sample, we substi- and +60◦. The letter A indicates the latitudinal extent− 0 0 of the streamer. The open magnetic field has positive tuted directly Bi Bi+1. The system was again let free to relax and attain7−→ a new steady state for each itera- polarity in the northern coronal hole and negative polar- tion i. The stability of the relaxation method was tested ity in the southern coronal hole. The streamer itself is so that we could be sure that each relaxed steady-state divided into four magnetic connectivity regions around solution did not depend on the history. That is, we found one X-type null point. The null point itself is located no hysteresis when cycling back and forth through dif- over the equator at r = 1.4 R⊙. The four connectiv- ferent states (given that we let the system relax at each ity regions mentioned above are then the group of small stage). Different permutations of the original time-series equatorial loop arcades place below the null point, the i =0, 1,...,N also lead to the exact same steady-states group of larger equatorial arcades above the null and as long as each solution’s set of parameters was kept the filling up most of the streamer, and the two groups of ar- same. The wind solution for each instant of time i de- cades to the north and south of the null (note that Figure 0 3 only shows one hemisphere, and that the system is sym- pends only on the corresponding Bi . Some of the runs were performed at different grid res- metric with respect to the equator). As the solar cycle olutions, and numerical convergence was verified. starts moving away from the minimum, new flux con- Both numerical codes are time-dependent in nature, centrations emerge at mid-latitudes (showing up as new but the result of the whole coupling procedure is a se- groups of coronal loops inside the equatorial streamer). quence of steady-state solutions. Our procedure gener- These new structures slowly migrate polewards, attain- ates a map of the poloidal coronal magnetic field and ing the streamer boundaries at about t = 2.5 yr. The wind flow during an activity cycle rather than the dy- equatorial streamer is disrupted at this point. A fraction namical evolution of a particular event. of the magnetic flux will remain in the equatorial region, Toroidal fields and flows will not be considered forming a smaller streamer (Figure 3, second panel, t2). in this paper. Concerning rotation, the sun is a The rest of the magnetic flux reconnects and forms new slow magnetic rotator, and therefore the magneto- plume/pseudo-streamer structures at mid-latitudes (as rotational effects on the poloidal wind flow are negligible in Pinto et al. 2010 and Wang et al. 2007). The letter (Belcher & MacGregor 1976). B indicates the pseudo-streamer position. New coronal 3. holes now appear at low latitudes. The polewards pro- CORONAL MAGNETIC FIELD gression continues at a steady pace opening up its way The temporal evolution of the corona and solar wind in by reconnecting with the open magnetic flux. At about response to the dynamo field variations is shown in Fig- t = 3.5 yr one of the magnetic arcades of each of the ure 3. Only the first 4 R⊙ of the northern hemisphere newly formed pseudo-streamer breaks and quickly opens are displayed at six different instants of the activity cy- up (Figure 3, third panel, t3, letter C). As a result, a cle, corresponding to the t1, t2, ..., t6 lines in Figure 2. new coronal hole with inverse magnetic polarity rapidly That is: t =0, 3.3, 3.6, 3.8, 4.4, 11 years. The colour- forms. This coronal hole will grow wider and fill up all scale represents the solar wind’s velocity projected on the the polar regions as the polarity inversions proceeds. The unit magnetic field vector in units of Mach number, that previous coronal hole’s field then closes down (Figure 3, is v B/(c B ). Orange/yellow and blue/green shades fourth panel, t ), and ends up disappearing below the · sk k 4 therefore trace different B-field polarities in the open surface (Figure 3, fifth panel, t5). The corresponding field regions (respectively, u B positive and negative). coronal arcades (the ones closing down near the poles) The sharp transitions between· positive and negative po- are indicated by the letter D in the figure. At t = 11 yr larities in this figure outline current sheets (note that the all traces of the previous coronal hole magnetic field have wind flow does not change sign across these transitions, disappeared (Figure 3, last panel, t6). The system is back but Br does). to a state very close to its original state, but with the po- Some elements and characteristics are observed consis- larity of the magnetic field reversed (the dynamo model tently throughout the whole cycle. Higher concentrations used here produces very regular and symmetric cycles). of magnetic flux at the surface (or equivalently, of cur- Note that the polarity reversal happens quickly in the rent below the surface) translate into coronal loop arcade corona, even if the underlying B0-field evolves slowly. systems. Strong flux concentrations appearing in coronal The wind flow is responsible for the quick “opening-up” holes shape up as helmet streamers, and end in a cur- of field lines as the magnetic flux concentrations evolve rent sheet which extends outwards. Smaller flux concen- slowly at the surface, and ultimately for the change of trations embedded in unipolar flux regions form nearly connectivity between contiguous regions. Furthermore, symmetric bipolar structures with no current sheet (as at the polar axis, the magnetic field’s inversion occurs at Coupling the solar dynamo and the corona 7

4 4

3 3

2 2

1.50

1 1

& % t = 0.0 yr t = 3.3 yr 1.00 0 1 0 2 0 1 2 3 4 0 1 2 3 4

4 4

0.50

3 3

0.00 2 2

1 1

-0.50

t = 3.6 yr t = 3.8 yr 0 3 0 4 0 1 2 3 4 0 1 2 3 4

4 4 -1.00

3 3

-1.50

2 2

1 1 !

t = 4.4 yr t = 11.0 yr 0 5 0 6 0 1 2 3 4 0 1 2 3 4

Figure 3. Snapshots of the evolution of the corona during the solar cycle (only the first ∼ 4 R⊙ and northern hemisphere are shown) at u B 1 6 · the instants t to t shown in Figure 2. White lines are magnetic field lines. The colorscale represents the quantity cskBk , that is, the wind flow velocity projected onto the signed magnetic field in units of Mach number. This quantity traces the B-field’s polarity in the open field regions. Red/orange means positive polarity, while green/blue means negative polarity. The large arrowheads show the local B-field orientation. The grey contour shows the sonic surface. The letters A, B, C and D indicate the positions of particular magnetic structures which we refer to in the text. 8 Pinto et al.

10 2 t1=0 yr t1=0 yr 5

sun 1 t2 t2 t3 0 t ,t 0 3 4 t4

r=15 R t

2 5 North Pole t5 2 r r r

r t5b=5.5 yr -5 B -1 t =5.5 yr B 5b

t6=11 yr -2 -10 t6=11 yr 1.0 1.5 2.0 2.5 3.0 -50 0 50 R [Rsun] latitude

6 10 t1=0 yr 4 5 r=1 R sun sun 2 r=15 R 0 sun 0 r~3 R 2 North Pole 2 r r r

r -5 -2 B

B r=1.1 Rsun

r=1.2 Rsun -4 -10 r=1.4 Rsun t =11 yr -6 6 0 2 4 6 8 10 -50 0 50 time [yr] latitude Figure 4. Top panel: Polar magnetic field as a function of radius at different moments of the cycle. The instants represented are the same as in the panels in Figure 3 (t1 to t6) plus an additional 40 2 t5b = 5.5 yr for completeness. Bottom panel: Brr at the north t1=0 yr pole as a function of time. B is in units of G and r is in units of 2 R⊙. Each curve corresponds to a different height. The r factor 20 accounts for the field’s decay due to a purely radial expansion (note sun 2 that Brr decays faster in the lower part of the domain, but not above). The polarity inversion at the surface is delayed with respect 0 r=1 R to higher coronal heights. This is due to the slowly progressing 2 r r

closing down of polar fields, as seen in Figure 3. B -20 different times at different heights. To better describe t6=11 yr this property, the top panel in Figure 4 shows radial cuts -40 of the polar magnetic field at the same moments as those -50 0 50 shown in Figure 3. The bottom panel shows the temporal latitude 2 evolution of the polar field at different heights. The Br Figure 5. Brr as a function of latitude at different heights, and sign-switch happens first at higher altitudes and proceeds at different moments of the cycle. The r2 factor accounts for the downwards. The downwards progression of the reversal field’s decay due to a purely radial expansion, as in the previous figure. B is in units of G and r is in units of R⊙. The instants of Br is quick (δt of order of a few days) between r = represented are the same as in the panels in Figure 3 (t1 to t6) 15 R⊙ and r = 2 R⊙, but from there on it slows down plus an additional t5b = 5.5 yr for completeness. Faraway from the considerably. The delay is δt 6 months between r = sun, the radial magnetic field is mostly uniform in latitude (except around current sheets, where it changes sign), independently of the 2 R⊙ and r = 1.6 R⊙, and≈ about 1 yr between r = complexity of the surface field. 1.6 R⊙ and r = 1.3 R⊙. The overall delay (between top and bottom) is about 4 yr. This delay in the lower The multiple current sheets shown in the intermediate coronal layers is due to the slow disappearance of the panels in Figure 3 may be interpreted as a highly warped coronal arcades near the poles, as shown in the last three current sheet in the real non-axisymmetric corona. panels of Figure 3. At the activity minimum the open flux is restricted ◦ Figure 5 shows Br as a function of θ at different al- to large polar coronal holes (about 30 in latitude in titudes and at different moments of the cycle. Faraway each hemisphere). This open flux will eventually fill all ◦ from the surface, B Brer and is nearly independent the available space at greater heights (spanning 180 in of the latitude, except≈ in the vicinity of a current sheet. latitude, from pole to pole), as the streamer thins out; Close to the surface (beneath r 2 3 R⊙), the mag- above r 3 R⊙ all the magnetic field is open. Closer to netic field organises in much more≈ complex− ways. The the maximum,≈ the open flux sources are more spread in potential component of the magnetic field dominates in latitude but cover a smaller latitudinal extent altogether the lower layers of the corona whereas the induced com- (about 5◦ in latitude in each hemisphere). These multiple ponent largely dominates above (as the wind flow be- thin coronal holes will nevertheless grow with height, and comes stronger and approaches its asymptotic velocity). will end up filling up all available space. Above r From there on, the solar wind flow and the total magnetic 2.5 R⊙ all the magnetic field is open. In other words, the≈ field essentially align with the radial direction. average flux-tube expansion factors will be much higher Coupling the solar dynamo and the corona 9 at the maximum than at the minimum. Figure 6 shows the latitudinal distribution of the solar wind speed at r =15 R⊙ (the domain’s outer bound- 4. SOLAR WIND SPEED ary) during the solar cycle. The wind velocity vs. latitude diagram shows good qualitative agreement with the Ur [km/s]; r = 15 R_sun predictions made by Wang & Sheeley (2006) matching t ULYSSES in situ measurements and the more recent 6 450. 10 multi-station IPS (interplanetary scintillation) coronal observations by Tokumaru et al. (2010). The separation between fast and slow wind is well visible in the figure. 430. At the activity minima, fast solar wind originates essen- 8 tially from high latitude regions, while the slow wind flows mostly closer to the streamer frontiers, at lower 410. latitudes. As the activity cycle progresses from the min- 6 imum to the maximum, the slow wind expands over towards the poles and takes over most of the latitudinal time [yr] t5 390. t domain. On the declining phase of the activity cycle, the 4 t 4 t3 fast wind recovers the polar regions and progressively ex- 2 tends towards lower latitudes, restraining the slow wind 370. flow to the equatorial region. Some irregularities appear 2 above this otherwise too simple scenario. Most remark- ably, two short-lived wind channels appear at low lati- 350. tudes between t = 3 yr and t = 4 yr, that is, during the 0 t1 -50 0 50 polarity inversion. This can be seen in Figure 3 (pan- latitude els t2, t2, t3); note how the blue shaded wind channel Figure 6. Solar wind speed at the outer boundary during the cy- appears and evolves. These flows originate in the newly cle as a function of time and latitude. Most of the time, faster wind formed mid-latitude coronal holes. They follow the corre- outflows occupy a large latitudinal extent ranging from the poles sponding magnetic flux-tubes, bending over from 50◦ to latitudes as low as ±20◦. The exception is the polarity reversal at the surface down to 30◦ at the outer boundary.∼ phase (t ∈ [2, 5] yr). Two low-latitude wind channels appear at ∼ about t = 3 yr, corresponding to the newly formed coronal holes Although they seem to fade away quickly in Figure 6, (see Figure 3, at t3). they actually last till the declining phase of the activity cycle, but the wind flowing within these coronal holes slows down. This slowing down is due to the increas- We focus now on the variation of the solar wind veloc- ingly higher expansion factor for the coronal hole, as can ity during the solar cycle, and on how it is distributed in be seen in Figure 3. Note how the blue shaded coronal latitude. The evolution of the coronal magnetic topology hole expands to fill the whole hemisphere faraway from ( 3) has a direct influence on the size and distribution of the sun while its latitudinal extent at the surface remains wind§ sources at the surface of sun, via changes in posi- approximately constant (from t3 to t5). This continues tion and width. The local magnetic pressure dominates while the polar closed-field regions progressively disap- over the wind’s dynamical pressure at the lowest layers, pears; all open flux will merge into a wide polar coronal and the local B-field’s amplitude and inclination mostly hole afterwards (t6). Also, the closing down of magnetic determine whether a given field line is open or closed. flux near the poles (during the cycle’s decay phase, at Conversely, the latitudinal distribution of the solar wind about t = 7 yr) is related to the appearance of fast wind faraway from the sun cannot be trivially predicted from flows close to the polar axis. These correspond to newly the magnetic field’s configuration at the surface. The so- formed thin polar flux-tubes expanding almost radially, lar wind flows accelerates along open field lines. A fluid which will also merge afterwards into the wide polar coro- element of cross-section A0 at the surface will accelerate nal hole. along a magnetic flux-tube with cross-section A (r). The The wind’s velocity values presented here are expected final velocity profile depends on the flux-tube’s expan- in situ A(r) to be lower than the values measured near the sion factor A0 . The expansion factor itself results from Earth’s orbit. The reader should note that, on one hand, the competition between the wind’s dynamical pressure our numerical domain extends only up to 15 R⊙, and and the magnetic pressure in the corona. that the wind flow has not yet reached its asymptotic ve- The terminal wind speed and flux tube expansion locity at this height. Nevertheless, the relative variations factor are inversely correlated at all latitudes and at of Vr in latitude should not change considerably. The all times, agreeing with the well established Wang– wind undergoes a purely spherical expansion between Sheeley–Arge semi-empirical relation (Wang & Sheeley 15 R⊙ and 1 AU. On the other hand, this does not fully 1990; Arge & Pizzo 2000). In the initial acceleration accounts for the low velocities values found, though. We phase (below r 3 R⊙) the expansion factors vary con- could expect velocities in the order of 500 600 km/s from siderably, both∼ in latitude and radius. At larger radii, this model if the domain extended up to− 1 AU. Faster though, the multiple and initially independent wind flows wind velocities require a complete and consistent ener- −2 merge into a bulk spherical outflow (A (r)/A0 r , getic treatment, which is beyond the scope of the current independently of the latitude). Latitudinal inhomo-∝ paper. Besides, Pinto et al. (2009) show that the anti- geneities still subsist in this flow, being a result of the correlation between terminal wind velocity and flux-tube evolution (acceleration) of each parcel of wind flow along expansion factor is still verified in self-consistent non- its path starting from the sun’s surface. isothermal cases. The only strong constraint over the 10 Pinto et al. domain’s radial extent regarding the physical correctness 15 15 of the numerical model is that it has to completely con- tain all critical surfaces (sonic and alfvénic; cf. 2.2), which it always does. § 10 10

5. MASS FLUX, MOMENTUM FLUX AND MAGNETIC BRAKING TORQUE 5 5

10 10 0 0

-5 -5

5 5

-10 -10

0 0 -15 t1 = 0.0 yr -15 t2 = 3.3 yr 0 5 10 15 0 5 10 15 Figure 9. Alfvén surfaces (black contours) at the same instants as in fig. 7 (instants t1 and t2). White lines are magnetic field lines, -5 -5 and the colorscale represents the wind’s poloidal Mach number. degree of axi-symmetry in the distribution of the streamers and coronal holes. These low-latitude wind streams -10 t = 0.0 yr -10 t = 3.3 yr correspond to the two channels visible in Figure 6, at 1 2 about t = 3 yr. 0 2 4 6 8 10 0 2 4 6 8 10 2 Figure 8 shows the total mass loss rate Figure 7. Mass flux ρVrr sin θ in the meridional plane at t = 0yr (left) and at t = 3.3 yr, about the polarity inversion (right), that π 2 ˙ 2 is, respectively, instants t1 and t2. The factor r sin θ is due to M =2πR0 ρVr sin θdθ (6) the spherical expansion of a surface element normal to the radial Z0 direction. Outflows which originate at lower latitude dominate the global mass loss rate. evaluated at the outer boundary of the numerical domain. The mass loss rate evolves in par with the activity cycle. That is, M˙ is maximal at about t = 3 yr (during the activity maximum) and minimal during the activity

-14 minima. The amplitude of the mass loss rate varies by 7·10 −14 a factor of about 1.6 in time, from 4.2 10 M⊙/yr −14× at the activity minimum and 6.9 10 M⊙/yr at the /yr] -14 × sun 6·10 activity maximum. This trend supports the idea that the lower latitude outflows — which appear mainly close 5·10-14 to the activity maximum — contribute with higher net mass outflow rates, as discussed in the previous para- d/dt M [M graph. Note that the increase in M˙ cannot be due to the 4·10-14 variations in the velocity of the wind, as it is actually 0 2 4 6 8 10 lower at almost all latitudes during the activity maxima time [yr] (see Figure 6) and contributes to lowering the net mass flux. On the other hand, the variations of ρ at the lower Figure 8. Mass loss rate M˙ during the solar cycle. numerical boundary are small throughout all the cycle. The amplitude of the density fluctuations at the surface Figure 7 shows the radial mass flux associated with within open-field regions is always below 6% of its aver- the solar wind in the corona at two different instants age value (amounting to at most 10% if both open and of the solar cycle. In both cases, the net mass outflow closed-field regions are considered altogether). in the polar coronal holes is higher near the streamer The way the open flux maps from the solar surface up boundaries than closer to the poles. At the maximum the outer domain must therefore be the main cause for of activity, thin coronal holes appear also at low lati- the variations found in the mass loss rate. tudes (as described previously). Despite their small lati- The solar wind outflow carries angular momentum tudinal extent at the surface, the associated mass flux is away from the sun. The specific angular momentum flux, important when compared to that in the polar coronal magnetic braking torque and spin-down time-scale are holes. For an outflow with a given latitudinal extent δθ, deduced from the Alfvén surface’s geometry and solar 2 the actual surface area it crosses is equal to 2πr sin θδθ. rotation rate Ω0. Therefore, low latitude coronal hole are more prone to The Alfvén surface is the geometric locus where the produce higher mass outflow rates. Arguably, this is a wind velocity equals the Alfvén speed cA = B/√4πρ. consequence of the axi-symmetrical nature of our model. The Alfvén radius rA is the cylindrical distance from Nevertheless, the real three-dimensional sun shows some the rotation axis to this surface. Classical wind theory Coupling the solar dynamo and the corona 11

momentum ﬂux rate is then 8 2

] l =Ω0 rA . (7)

sun h i 6 The resulting torque applied on the sun is

> [R (Matt & Pudritz 2008) alfven 4 τ = M˙ Ω r2 . (8) < R − 0h Ai 2 The angular momentum per unit volume J of a parcel of solar wind plasma rotating with azimuthal velocity vφ 0 2 4 6 8 10 is time [yr] Jw = ρr sin θvφ . (9)

1.4·1018 The angular momentum per unit volume crossing a ˙ 1.2·1018 surface element dA is then Jw = JwvrdA. Integrating over a spherical surface of radius r0 and assuming axi- 1.0·1018 symmetry translates into 8.0·1017 π ˙ 3 2 6.0·1017 Jw =2πr0 ρvrvφ sin θdθ . (10) 0 4.0·1017 Z We then deﬁne the magnetic spin-down time-scale as 2.0·1017

Angular momentum flux [cgs] J⊙ δtsd = , (11) 0 2 4 6 8 10 J˙w time [yr] where J⊙ is the sun’s angular momentum. We estimated

R⊙ 8π 4 48 2 −1 J⊙ = Ω0 ρ (r) r dr 1.84 10 g cm s 1012 3 0 ≈ × Z (12) (Gilman et al. 1989; Stix 2002) using a seismically cal- ibrated solar model for ρ (r) (a CESAM model, Morel 1011 1997; Brun et al. 2002). The main difficulty now lies in the definition of rA . Figure 9 shows the Alfvén surface at two differenth in-i Spin-down timescale [yr] 10 stants of the cycle (t = 0, 3 yr). This critical surface 10 0 2 4 6 8 10 shows a regular shape for most of the activity cycle, be- time [yr] ing close to spherical at most latitudes (especially, higher latitudes). At low latitudes, though, the Alfvén surface Figure 10. Mean Alfvén radius hrAi (top), specific angular mo- 2 approaches a more cylindrical shape. Some irregulari- mentum flux ΩhrAi (middle) and magnetic braking time-scale ˙ ties appear as inward incursions as B vanishes in current δtsd = J⊙/J (bottom) during the solar cycle. J⊙ is the sun’s sheets (there’s a small but finite outflow, so v/c angular momentum (Gilman et al. 1989); J˙ is the wind’s angular A → ∞ momentum loss rate. there). The most evident example of such an excursion corresponds to the equatorial streamer (e.g Figure 9, left panel). We defined here the average Alfvén radius rA as the average cylindrical radii of the Alfvén surfaceh (thati (Weber & Davis 1967, and many others) states that the is, rA, the distance to the axis) weighted by the local angular momentum balance problem can be simplified mass flux r2 sin θρv crossing the surface as follows. The plasma inside the Alfvén surface is kept in solid rotation while flowing outwards along magnetic r2 sin θρv r (θ) nˆdθ r = · A . (13) field lines. The plasma flow then becomes super-alfvénic h Ai r2 sin θ ρv dθ and its angular momentum is conserved thereafter. The R k k physical process responsible for maintaining the solid ro- The sections of the AlfvénR surface crossing heliospheric tation while V

Varying |B(t=0)| suggested by Matt & Pudritz (2008), where B is the 10 global field’s amplitude. This scalar quantityk is wellk de- t 1 fined for any given analytical external magnetic field but not for the ones obtained numerically from the dynamo model. We chose here to associate it to the unsigned sun magnetic flux integrated over the solar surface (after > / R

a comparing both deﬁnitions for the dipolar test cases).

sun and t = 5 yr), drawing a closed cycle in the diagram. This result suggests that the Alfv´en radius correlates > / R a positively with B 2/M˙ when the dipolar component of

t1 coronal phenomena is very large. The second and perhaps more important justiﬁcation is that varying the potential magnetic ﬁeld in time introduces a non causal

sun perturbation to the system. That would correspond to t5 an instantaneous propagation of a perturbation to the > / R a t2 ﬁeld sources. This violation of causality remains even

t3 t4 for a continuous and arbitrarily slow evolution of the background potential field. Note that localised pertur- bations to a MHD system are propagated away with 1 phase velocities which correspond to the MHD wave 102 103 104 105 2 2 modes. These can be arbitrarily small (even null in some B R / M sun dot places) and very anisotropic. To work around this is- Figure 11. Mean Alfvén radius as a function of Υ. Top: variation sue one would need to include the field sources in the of the amplitude of the external magnetic field for our t1 = 0 yr domain (and then self-consistently compute the system’s case. Middle: dipolar benchmark case (varations of the external field’s amplitude, as above). Bottom: the same relation, but response to their perturbation). Alternatively, one could throughout the cycle. The labels t1 to t6 identify the instants emulate the proper physical behaviour of such a system shown in Figure 3. The label t1 appears both on the top and by injecting/propagating the hydro-magnetic perturba- bottom panels, as it corresponds to the same state. tions through the numerical boundaries. This issue will be the subject of future work. test cases in which we varied only the external magnetic 7. SUMMARY AND CONCLUSIONS field’s amplitude, and not its topology (i.e, the potential We have performed MHD numerical simulations cou- magnetic field only varied by a multiplicative factor be- pling a solar dynamo model with a corona and solar wind tween consecutive runs). In the first case, we took the model throughout a complete activity cycle. In short, we first element of our time-series (t = 0 in the figures in found that: the preceding sections) and varied the amplitude of the external field by a constant multiplicative factor. The The latitudinal distribution of the asymptotic wind second case corresponds to a purely dipolar external field • velocities is sensitive to the magnetic topology as B0 whose amplitude we also varied, letting us perform it varies during the solar cycle. The fast wind - a comparison with previous studies. We then computed slow wind pattern shows good qualitative agree- the mean Alfvén radius rA and the mass loss rate M˙ ment with those in Wang & Sheeley (2006) and for each new run and testedh i against the correlation Tokumaru et al. (2010), as shown in Figure 6. m 2 The polarity reversal happens rather abruptly in rA m ( B R⊙) h i Υ = k k • the corona, in contrast with the progressive evolu- R⊙ ∝ " M˙ # 14 Pinto et al.

tion of the solar wind’s velocities and of the surface Gilman, P. A. 1983, The Astrophysical Journal Supplement magnetic field (Figure 4). Series, 53, 243 Gilman, P. A., Morrow, C. A., & Deluca, E. E. 1989, The Sun’s global mass loss rate, Alfvén radius and mo- Astrophysical Journal, 338, 528 • Glatzmaier, G. A. 1985, Geophysical and Astrophysical Fluid mentum flux all vary considerably throughout the Dynamics, 31, 137 cycle (Figures 8 and 10). The dominant causes are Grappin, R., Aulanier, G., & Pinto, R. 2008, Astronomy and the position and latitudinal extent of the photo- Astrophysics, 490, 353 spheric sources of solar wind and the geometry of Grappin, R., & Léorat, J. 2001, Astronomy and Astrophysics, the Alfvén surface. 365, 228 Grappin, R., Léorat, J., & Buttighoffer, A. 2000, Astronomy and The zones of application of the braking torque due Astrophysics, 362, 342 • Grappin, R., Leorat, J., Cavillier, E., & Prigent, G. 1997, to the wind vary in time. Overall, the wind’s break- Astronomy and Astrophysics, 317, L31 ing torque should contribute to slow down the sur- Grappin, R., Wang, Y., & Pantellini, F. 2011, The Astrophysical face layers at high latitudes, but regions of appli- Journal, 727, 30 cation of torque appear occasionally at lower lati- Guerrero, G., & de Gouveia Dal Pino, E. M. 2008, Astronomy tudes. and Astrophysics, 485, 267 Guhathakurta, M., Sittler, E. C., & Ofman, L. 2006, Journal of Geophysical Research (Space Physics), 111, 11215 Future work will focus on testing other types of solar Hammer, R. 1982, Astrophysical Journal, 259, 767 dynamo models, using Babcock-Leighton flux transport Hansteen, V. H., & Leer, E. 1995, Journal of Geophysical based on large scale meridional circulation (Dikpati et al. Research, 100, 21577 2004; Jouve & Brun 2007) or turbulent magnetic pump- Hu, Y. Q., Feng, X. S., Wu, S. T., & Song, W. 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