Production of sunspots and their effects on the corona and solar wind: Insights from a new 3D flux-transport dynamo model
Rohit Kumar, Laurène Jouve, Rui F. Pinto, Alexis P. Rouillard
PProduction of sunspots and their effects onthe corona and solar wind: Insights from anew 3D flux-transport dynamo model
Rohit Kumar , ∗ , Laur `ene Jouve , Rui F. Pinto and Alexis P. Rouillard Institut de Recherche en Astrophysique et Plan ´etologie, Universit ´e de Toulouse,CNRS, UPS, CNES, 31400 Toulouse, France
Correspondence*:Rohit [email protected]
ABSTRACT
We present a three-dimensional numerical model for the generation and evolution of the magneticfield in the solar convection zone, in which sunspots are produced and contribute to the cyclicreversal of the large-scale magnetic field. We then assess the impact of this dynamo-generatedfield on the structure of the solar corona and solar wind. This model solves the inductionequation in which the velocity field is prescribed. This velocity field is a combination of a solar-likedifferential rotation and meridional circulation. We develop an algorithm that enables the magneticflux produced in the interior to be buoyantly transported towards the surface to produce bipolarspots. We find that those tilted bipolar magnetic regions contain a sufficient amount of fluxto periodically reverse the polar magnetic field and sustain dynamo action. We then track theevolution of these magnetic features at the surface during a few consecutive magnetic cycles andanalyze their effects on the topology of the corona and on properties of the solar wind (distributionof streamers and coronal holes, and of slow and fast wind streams) in connection with currentobservations of the Sun.
Keywords: solar magnetic cycle, mean field dynamo, flux-transport, sunspots, field reversal, solar corona
The Sun’s magnetism is responsible, among other things, for the production of sunspots and polar fieldreversals [1]; which result in coronal heating and solar wind. This activity is connected to what is nowcurrently called space weather and affects the Earth’s environment, in particular satellite operations andtelecommunications.Among the different dynamo models applied to the Sun’s magnetic field, one has been particularlysuccessful at reproducing many of the observed solar features: the Babcock-Leighton (BL) dynamo model,proposed in the 60’s by Babcock and Leighton [2, 3]. In this model, the toroidal field is generated nearthe base of the solar convection zone (SCZ) by the shearing induced by the differential rotation. Thistoroidal field then gets transported, due to the magnetic buoyancy, to the solar surface to produce bipolarmagnetic regions (BMRs or sunspots) [4]. Because of the Coriolis force acting on the rising toroidalstructures, the emerged sunspot pairs possess a tilt with respect to the East-West direction such that the a r X i v : . [ a s t r o - ph . S R ] J a n umar et al. Production of sunspots and their effects on the corona and solar wind leading spot is located at a lower latitude than the trailing one [5]. These tilted BMRs are then advectedby the surface flows and are subject to the turbulent diffusion induced by small-scale convective motions.If some trans-equatorial cancellation is allowed for the leading spots of both hemispheres, the net fluxadvected towards the poles is then of one particular polarity, opposite to the one of the polar field. Thisnew flux will thus reverse the polar field (this is precisely the BL mechanism) which in turn will producethe toroidal field of the next cycle. If the generation of the toroidal field through differential rotation iswell accepted, the crucial role of spots to reverse the poloidal field is more debated. However, a recentstudy by Dasi-Espuig et al. [6] in 2010 has shown that a correlation indeed exists between an observablemeasurement of the Babcock-Leighton mechanism and the strength of the next solar cycle. This wouldindicate that the sunspots indeed play an important part in the dynamo cycle.The solar dynamo has been studied for decades using two different approaches: either two-dimensional(2D) kinematic mean-field dynamo models where the evolution of the large-scale magnetic field is computedfor a prescribed velocity field [7, 8], or three-dimensional (3D) global models where the evolution ofboth the velocity and the magnetic fields are studied by solving the full set of magnetohydrodynamic(MHD) equations in which the velocity and magnetic fields interact nonlinearly (see reviews by Mieschand Toomre [9] and Brun et al. [10]). Several 2D BL flux-transport dynamo models have been used tostudy the solar magnetism [11, 12]. But, in those 2D models, it is assumed that the BL process is the sourcefor the poloidal field and thus a source term is added to the poloidal field equation in an ad hoc way. In 3Dmodels, the strong toroidal structures built in rapidly-rotating stars can become buoyant [13, 14] but rarelyrise all the way to the top of the computational domain and, consequently, those models do not producespots. It is thus not possible in such “spotless” models to assess the potential role of spots in the large-scalefield reversals and to study the impact of such evolving spots on the coronal structure.We propose here to develop a 3D kinematic solar dynamo model in which the toroidal field owes its originto the shearing of the poloidal field by the differential rotation and where an additional buoyancy algorithmis implemented to force strong toroidal regions to rise through the convection zone and produce spots. Thefirst objective here is to see the BL mechanism at play, i.e., study the ability of these tilted BMRs to reversethe polar field and serve as a seed for the next toroidal field. In this first step, we will not try to particularlycalibrate our model on real solar observations but this model, strongly relying on surface features likesunspots, will serve as a first step towards introducing data assimilation for long-term forecasting. Similarmodels have been developed recently, using two approaches which differ in the way BMRs are producedat the surface. Miesch and Dikpati [15], followed by Miesch and Teweldebirhan [16], used a double-ringalgorithm, putting BMRs directly at the solar surface and extracting an equivalent toroidal flux at thebase of the convection zone. On the other hand, Yeates and Mu˜noz-Jaramillo [17] used a more realisticmethod in which an additional velocity field is responsible for the transport of toroidal flux from the baseof convection zone to the solar surface. Miesch and Teweldebirhan [16] indeed obtained cyclic reversals ofthe magnetic field using their “spot-maker” algorithm where the step of flux rising from the bottom of theconvection zone is not explicitly modeled. In the more realistic model of Yeates and Mu˜noz-Jaramillo [17],only one solar cycle was modeled and the possibility to have a self-sustained dynamo was not studied.We here describe a model similar to Yeates and Mu ˜noz-Jaramillo [17] but where several cycles arecomputed. We then reconstruct the evolution of the coronal magnetic field in response to the modeleddynamo field and study how coronal holes vary in size and position and how the open flux at variouslatitudes evolve along the magnetic cycle. Those quantities are particularly important to assess the 3Dstructure of the solar wind and how the wind speed varies during a full magnetic cycle. We note that weare not trying here to simulate the whole complexity of the solar magnetic field evolution from the base
This is a provisional file, not the final typeset article umar et al. Production of sunspots and their effects on the corona and solar wind of the convection zone to the corona and solar wind structure, we are instead showing a proof-of-conceptthat important large-scale features can be reproduced by our simplified models and that, thanks to thosesimplifications and the small computational cost, data assimilation could be easily introduced in thisintegrated model to help forecasting the future evolution of the large-scale solar magnetic field.The rest of the paper is organized as follows: we describe the details of our new 3D kinematic dynamomodel in Sec. 2. In Sec. 3, we present the simulation of the self-sustained Babcock-Leighton dynamo. InSec. 4, we focus on the coronal magnetic topology and the solar wind during a full solar cycle. Finally, inSec. 5, we summarize our results.
We solve the magnetic induction equation in a 3D spherical shell. This equation reads: ∂ B ∂t = ∇ × ( v × B ) − ∇ × ( η ∇ × B ) , (1)where B is the magnetic field, v is the prescribed velocity field, and η is the magnetic diffusivity. In oursimulations, the prescribed velocity field is a combined effect of the differential rotation and the meridionalcirculation.We perform dynamo simulations in a spherical shell geometry using the pseudo-spectral solver MagIC [18,19]. MagIC employs a spherical harmonic decomposition in the azimuthal and latitudinal directions, andChebyshev polynomials in the radial direction. For time-stepping, it uses a semi-implicit Crank-Nicolsonscheme for the linear terms and the Adams-Bashforth method for the nonlinear terms.The inner and outer radii of the computational shell are [0 . , . R (cid:12) , where R (cid:12) is the solar radius. Wechoose N r = 64 grid-points in the radial, N θ = 128 points in the latitudinal, and N φ = 256 points in thelongitudinal direction. Simulations are performed by considering insulating inner and outer boundaries forthe spherical shell. We employ an initial magnetic field which is the combination of a strong toroidal and arelatively weak poloidal magnetic field. The strong toroidal field is chosen so that the magnetic flux dueto the BMRs generated at the surface would be sufficient to reverse the polar field of the first cycle. Wethus make sure that the initial conditions are favorable to produce a first reversal. We however note that thesteady-state solutions are not very sensitive to our choice of initial conditions. In this section, we present the ingredients of the dynamo model we used, in particular the velocity fieldand the magnetic diffusivity profile.We prescribe the velocity field such that the flow has a differential rotation similar to that observed in theSun through helioseismology [20]. In addition, the velocity field also consists of a meridional flow, whichis poleward near the surface and equatorward near the base of the convection zone. The meridional flow isan important ingredient since it is thought to be responsible for the advection of the effective magnetic fluxof the trailing spots towards the poles to reverse the polarities of the polar field [21, 22, 23, 24]. Both theexpressions of the differential rotation and the meridional flow are slightly modified versions of the onesused in [25].
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Production of sunspots and their effects on the corona and solar wind
In Fig. 1(a), we show the differential rotation superimposed with the meridional flow in our simulations.The rotation is strongest in the equatorial region, and it decreases as we move towards the poles. Theobservation of solar velocities suggest that the magnitude of the averaged meridional circulation atthe solar surface ( ≈ m/s ) is almost times smaller than the rotational velocity at the equator( ≈ km/s ) [26]. Assuming those observed values and choosing the solar radius as the characteristiclength-scale, the Reynolds numbers associated with the longitudinal velocity ( V φ = r sin θ Ω ) and withthe latitudinal velocity ( V θ ) are Re = V φ R (cid:12) /η s ≈ (maximum V φ at the surface near the equator) and Re p = V θ R (cid:12) /η s ≈ (maximum V θ at the surface), respectively, where η s is the value of the magneticdiffusivity at the surface. The rotation rate and the latitudinal velocity are illustrated in Fig. 1. (a) (b) Figure 1.
Plots of longitudinally averaged (a) differential rotation ( Ω ) superimposed with the meridionalflow (dashed lines: counter-clockwise, solid lines: clockwise) and (b) latitudinal velocity ( V θ ) in theconvection zone. Here Ω and V θ are in non-dimensional units. In dimensional units, V φ (= r sin θ Ω) ≈ km/s at the surface near the equator and V θ ≈ m/s at the surface (latitude: ± ◦ ).The magnetic diffusivity is a two-step function similar to the one defined in Yeates and Mu˜noz-Jaramillo [17]. The only differences lie in the values chosen for the various diffusivities. The valueof the diffusivity is smallest near the stable radiative zone η c = 5 × cm .s − , larger in the bulk of theturbulent convection zone η = 10 cm .s − and the largest near the outer surface η s = 2 × cm .s − ,where the turbulent diffusion induced by small-scale convective motions is thought to be even moreenhanced.We now describe how the rise of toroidal flux through magnetic buoyancy was implemented in the code. Following Yeates and Mu˜noz-Jaramillo [17], we employ an additional velocity field which transportsthe magnetic flux produced in the solar interior towards the surface. This velocity models the effects ofmagnetic buoyancy which is responsible for the rise of strong magnetic field from the base of the convectionzone to the surface. This additional velocity contains two components: a radial one to model the rise and avortical one to produce a tilt. The radial velocity as a function of longitude and latitude ( φ, θ ) is describedas V r = V r exp (cid:34) − (cid:40)(cid:18) φ − ¯ φσ φ (cid:19) + (cid:18) θ − ¯ θσ θ (cid:19) (cid:41)(cid:35) , (2)where ( ¯ φ, ¯ θ ) corresponds to the apex of the rising flux tubes, V r is the amplitude of the velocity (thecorresponding Reynolds number is ), σ θ = σ φ = 5 degrees. In our model, ¯ θ and ¯ φ are randomly chosen, This is a provisional file, not the final typeset article umar et al. Production of sunspots and their effects on the corona and solar wind such that the positions of the emerging regions at the surface are random. We choose in our particularmodel to have emergence of 32 BMRs every 4.5 months. Taking into consideration the latitudes of theobserved sunspots, we choose the values of ¯ θ to stay between [ − , +35] degrees.For representing the effect of the Coriolis force due to the Sun’s rotation, we then incorporate anadditional vortical velocity, which imparts a tilt to the emerging magnetic flux ropes. The vortical velocityis a combination of latitudinal and longitudinal velocity components. The vortical velocity is tuned so thatthe tilt in the BMRs is clockwise in the northern hemisphere, and anti-clockwise in the southern hemisphereand corresponds to values observed at the solar surface (between ◦ and ◦ [27]). The vortical velocity isalso designed such that the tilt angle increases as we move towards the poles, following Joy’s law [28]. Wedo not give the expression of this vortical component as it is identical to the one described by Yeates andMu˜noz-Jaramillo [17].In agreement with the physics of magnetic buoyancy instabilities, the additional velocity field is appliedonly for a toroidal magnetic field B φ > B lφ . In dimensional units, B lφ ≈ × Gauss. Below this value,the field is thought to be strongly influenced by the Coriolis force and thus rise parallel to the rotationaxis [29]. At the other end, we also suppress the effects of the additional velocity field for B φ > B hφ .In dimensional units, B hφ ≈ . × Gauss. Indeed, when the field is too strong, it is assumed to risevery rapidly to the surface without being affected by the Coriolis force. As a consequence, the producedBMRs will not be tilted and will thus not take part in the reversal of the polar field since the longitudinallyaveraged net flux will be zero [5].
Time M a g n e t i c E n e r g y E pol E tor Time M a g n e t i c E n e r g y Figure 2.
The non-dimensional poloidal and toroidal magnetic energy with time (magnetic diffusion).Time t = 1 corresponds to one magnetic diffusion time (using the surface value for the magnetic diffusivity),which is approximately years. Frontiers 5 umar et al.
Production of sunspots and their effects on the corona and solar wind
Starting with the initial conditions described in Sec. 2.1, we follow the time evolution of the magneticfield components in our new 3D kinematic dynamo model.In Fig. 2, we show the time-evolution of the toroidal and poloidal components of the magnetic energy.We clearly see here an exponential growth of the poloidal magnetic field and then a saturation of bothcomponents, as expected in a self-sustained dynamo. As discussed in Sec. 2.3, magnetic buoyancy acts onthe toroidal field only when B lφ < B φ < B hφ , i.e., there is upper and lower cutoffs on the emerging toroidalflux. These cutoffs introduce an additional nonlinearity in the system, which causes the saturation of themagnetic energy with time. This nonlinearity plays a similar role as the quenching term in 2D kinematicdynamo models. The saturation level of the magnetic energy depends on both the upper and the lowercutoffs on B φ . Indeed, when more flux is allowed to reach the surface (i.e., when B lφ is lower and B hφ ishigher), the magnetic energy naturally saturates at a higher level. (b) - (c) (d) (a) Figure 3.
Snapshots of the longitudinally averaged toroidal magnetic field ( B φ ) at different stages duringhalf a solar magnetic cycle. Subfigures show the polarity reversals of the toroidal field. Here, the magneticfield is in a non-dimensional unit, a value of 1 corresponds to Gauss. (a) t=0.51 (c) (d) t=0.60 (b)
Figure 4.
Snapshots of the longitudinally averaged non-dimensional radial magnetic field ( B r ) at differentstages during half a solar magnetic cycle. Subfigures show the polarity reversals of the polar field. Here, aunit magnetic field corresponds to Gauss.Fig. 3 shows snapshots of the longitudinally averaged toroidal magnetic field during half a magneticcycle (the time duration between two consecutive polarity reversals). Initially, the toroidal field has a
This is a provisional file, not the final typeset article umar et al. Production of sunspots and their effects on the corona and solar wind positive (resp. negative) polarity in the northern (resp. southern) hemisphere, as seen on the first panel[Fig. 3(a)]. In Fig. 3 (b), a new negative polarity field starts to be amplified in the northern hemisphere andgets transported through advection by the meridional flow and diffusion inside the convection zone. At alater time, this newly generated toroidal field occupies the main part of the convection zone [Figs. 3 (c) and(d)] and will serve as a seed for the emergence of new BMRs of opposite polarity. In Fig. 4, we illustratethe snapshots of the longitudinally averaged radial magnetic field at the same times as in Fig. 3. We againclearly see the reversal of the polar field in time. At t = 0 . , the dominant toroidal field is positive in thenorthern hemisphere [Fig. 3(a)], creating BMRs with a positive trailing polarity at the surface. This positivepolarity is then advected towards the poles to reverse the negative polar field, as seen in Fig. 4(b). Whenthe negative toroidal field starts to be dominant [Fig. 3(c)], it is now BMRs with a negative trailing spotswhich will emerge, as seen in Fig. 4(c). It will then again start to cancel the positive polar field [Fig. 4(d)]and finally reverse it to start a new cycle. We note here that the typical values of the poloidal field at thepoles is very high (of the order of several kiloGauss) compared to real solar observations. This was alreadyobserved in a similar model in [16]. This is one of the shortcomings of our model. We know however thatthis amplitude will be strongly related to our buoyancy algorithm and for example reducing the numberof emerging spots or the frequency of emergence will reduce the polar field by the same amount. A fullparametric study and a calibration on the real Sun will be the subject of a future article. The buoyancy acting on the toroidal field near the base of the convection zone results in the production ofmultiple tilted BMRs at the surface. The distribution of radial magnetic field at the surface as a functionof time is shown in Fig. 5. The horizontal axes represent the longitude and the vertical axes the latitude.In this figure, we can clearly see the process through which the polar field reverses due to the net flux ofBMRs. Indeed, a new set of BMRs gets generated randomly at different longitudes and latitudes every 4.5months so that we have enough magnetic flux available at the surface in order to reverse the poloidal field.The net magnetic flux of the BMRs at the surface gets advected towards the poles due to the meridionalflow and magnetic diffusion, which leads to the polarity reversals of the previous poloidal field at the poles(see Fig. 5). Subsequently, the shearing of poloidal field due to the differential rotation will reproduce thetoroidal field at the base of the convection zone, completing the whole magnetic cycle in our model.As expected, the polarities of the produced BMRs change along with the polarity of the toroidal field nearthe base of the convection zone. As the polarity of the toroidal field reverses in the two hemispheres, thepolarities of the generated BMRs also reverse. In Fig. 5(a), we see the BMRs generated by a toroidal fieldof positive polarity in the northern hemisphere (positive trailing spot), whereas Fig. 5(d) demonstrates thatthe BMRs produced by a toroidal field of negative polarity now have a negative trailing spot. We see herethe BL mechanism at play, i.e., the ability of BMRs to be advected towards the poles to reverse the field ofthe previous cycle. In this model, BMRs thus play a crucial role in the cyclic behaviour of the magneticfield. Note however that in our model, only few bipolar spots appear at latitudes lower than ± degrees.This is another shortcoming of our model which is being studied at the moment. This can be adjustedby modifying the meridional circulation profile and speed at the base of the convection zone. Indeed, inthis current model, the new toroidal field gets generated at high latitudes and transported to the surface toproduce spots before getting significantly advected towards the equator by the meridional flow (see Fig. 3).Therefore, the strong toroidal field does not really reach low latitudes and thus spots are not produced atthose latitudes, contrary to what is observed in the Sun. By changing the meridional flow profile however, astronger advection of the toroidal field towards the equator can be achieved and more spots can emerge at Frontiers 7 umar et al.
Production of sunspots and their effects on the corona and solar wind low latitudes. At this stage though and as stated above, we are not trying to calibrate the model to real solarobservations, this will be the subject of future work. t=0.511000500-1000-5000 (a) L a t i t u d e t=0.601000500-1000-5000Longitude (d) -120 -60 0 120606030-30-600 t=0.541000500-1000-5000 (b) t=0.561000500-1000-5000 L a t i t u d e L a t i t u d e L a t i t u d e (c) Figure 5.
Snapshots of the non-dimensional radial magnetic field ( B r ) at the surface showing (a) thelarge-scale polar field along with tilted BMRs, (b) reversal of the polar field, (c) sunspot minimum, and(d) the polar field and the BMRs after the polarity reversals. Here, a unit magnetic field corresponds to Gauss.
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Time L a t i t u d e Figure 6.
Butterfly diagram for the mean radial magnetic field ( Br ) at the surface showing field reversals.The horizontal axis represents the magnetic diffusion time.To get an idea about the length of the magnetic cycle, we plot the longitudinally-averaged surface radialmagnetic field as a function of time and latitude, known as the butterfly diagram. This is illustrated inFig. 6. We note that since the BMRs are tilted, the longitudinally averaged flux at mid-latitudes is non-zero,the tilt is thus crucial for the BL mechanism to work since a net flux needs to be advected towards thepoles. The advection of this net flux is clearly visible in Fig. 6 above 40 degrees and all the way to thepoles where the previous polarity eventually reverses. In this case, the average time for the poloidal fieldreversal is almost years, which makes the length of the complete magnetic cycle equal to years. In thismodel, the length of the cycle is highly sensitive to the strength of the meridional circulation; it decreasesfor stronger meridional flow. We have not tried to calibrate our model on the Sun yet, but we know thatmodifying the meridional flow amplitude by some fraction, keeping in the range of values observed on theSun, will help us get closer to an 11-yr cycle. The formation and evolution of BMRs and the slower variation of the large-scale field both impactthe large-scale topology of the magnetic field of the solar corona. In order to analyze these effects, weextrapolated the surface magnetic field computed by the dynamo model to the solar corona by applying thePotential-Field Source-Surface (PFSS) method. We set a constant source-surface radius R SS = 2 . R (cid:12) forall the extrapolations and follow the extrapolation methods described by Wang and Sheeley [30].Fig. 7 shows a sequence of snapshots of the coronal magnetic field at times t = 0 . , . , . , . (as in the previous figures). The radial component of the magnetic field at the surface is represented ingray scale (between ± × Gauss), with black and white representing respectively the negative andpositive polarity. The field line colours indicate that the radial component of the coronal magnetic fieldis either positive (green) or negative (violet). During the rise phase of the cycle (first panel in the figure),emerging BMRs perturb the quasi-dipolar configuration of the background coronal field, composed mostlyof large trans-equatorial magnetic loop systems (streamers) and polar coronal holes which open up with
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Production of sunspots and their effects on the corona and solar wind height and end up filling up all the space above (expanding to all latitudes and longitudes at . R (cid:12) ). Thenewly emerged magnetic flux systems start by gently bending this quasi-bimodal distribution, makingstreamers incur into higher latitudes and coronal holes extend towards lower latitudes. This continuousdistribution of topological elements breaks up later on as the cycle proceeds, with new streamers appearingfrequently at mid latitudes at sunspot maximum (second panel) and other closed-field structures (such aspseudo-streamers) even appearing at the polar regions. After the global polarity reversal, these high-latitudeloop systems gently fade away, letting the corona relax again toward a simpler configuration until thefollowing minimum.Some BMRs produce a strong imprint on the overlying coronal fields even much after their emergence atthe surface. Shearing by differential rotation elongates the emerged flux in azimuth until they get diffusedaway. Strong enough BMRs produce long polarity inversion lines that last long enough to sustain looparcade systems that extend above them (e.g., in the two last panels, on the northern hemisphere, left side ofthe image). Overall, there seems to be a rather large fraction of closed to open magnetic flux between thesolar surface and the source-surface, with open flux regions being rooted on small portions of the surface.This is more easily visible in Fig. 8, which shows maps of the regions of the surface that are magneticallyconnected to the coronal holes that form above them. The colour scheme is the same as in Fig. 7. Incomparison to the real Sun, the total extent of sources of open field is small, especially the polar regions atsolar minimum. In fact, the emerging regions hold very intense and concentrated magnetic fluxes. Fig. 9shows the variation of the total unsigned open magnetic flux (measured at the source surface) as a functionof time for the whole duration of the dynamo simulation. The high and low latitude components of theopen flux (above and below ± ◦ in latitude) are also shown as blue and red lines, respectively. The totalopen flux is dominated by its high-latitude component through most of the solar cycle, the only exceptionsoccurring at sunspot maximum, close to polarity reversal. As expected, the high-latitude component isclearly anti-correlated with the sunspot cycle, while the low-latitude component shows no such clearcorrelation, being only marginally stronger at sunspot maximum (see Wang and Sheeley [31]). The variations of magnetic field geometry in the corona caused by the evolution of the solar dynamoare expected to perturb the large-scale properties of the solar wind flow. We have used the solar windmodel MULTI-VP (Pinto & Rouillard [32]) to analyze the response of the solar wind to these variations.We initiated the wind model by selecting a large collection of open magnetic flux-tubes from the PFSSextrapolations of the dynamo surface fields (see Sec. 4.1). The flux-tubes are seeded uniformly throughoutthe source-surface with an angular resolution close to ◦ (corresponding to more than flux-tubes persingle full map). The model computes a solar wind profile from the surface of the Sun up to r = 31 R (cid:12) foreach individual flux-tube, fully taking its geometry into account (field amplitude, expansion and inclinationprofiles). Finally, the one-dimensional solutions are reassembled into a spherical data-cube spanning thewhole solar atmosphere. Figure 10 shows a series of maps representing the wind speed and density at r = 21 . R (cid:12) , at two different instants of the simulation (close to sunspot maximum and minimum). Thewind speed is expressed in km/s and density in cm − . The green dashed lines represent the position of thepolarity inversion line in the corona above the source-surface, which indicates the shape and location of theheliospheric current sheet (HCS). Panel (a) of the figure shows a highly warped HCS, as can be found inthe solar atmosphere for periods close to sunspot maximum. Panel (b) shows a more moderately warpedHCS that is trying to relax to a configuration more typical of solar minima. This is a provisional file, not the final typeset article umar et al. Production of sunspots and their effects on the corona and solar wind
We obtain in all the cases a distribution of slow and fast solar wind streams (respectively below and above ∼
450 km / s ) that displays many of the main properties of the actual solar winds. The high-speed windstreams are mostly concentrated at high latitudes, being formed within polar coronal holes (see Fig. 8),with the highest wind speeds (reaching more than
750 km / s ) being attained during solar minimum. Somestreams of fast wind also appear at low latitudes during the phases of high solar activity. The transitionbetween slow and fast wind streams is for the most rather sharp (typically only a few degrees wide).However, the total angular extent of slow wind seems to be significantly larger than that of the actual solarwind. But that is consistent with the small coronal hole footpoint areas shown in Fig. 8 and discussed inSec. 4.1, which imply a prevalence of flux-tubes with very strong expansions between r = 1 and . R (cid:12) and that stretch over large azimuthal extents, implying that a significant number of wind flows are driventhrough highly inclined sections during their acceleration (see discussion by Pinto et al. [33]). The windmaps also indicate the presence of wind streams with very low speed ( ∼
200 km / s ) in these regions withhigh magnetic expansions and field-line inclinations which are consistent with the very slow wind streamsreported by Sanchez-Diaz et al. [34] (that should disappear at higher heliospheric altitudes). The winddensity is anti-correlated with the wind speed (with the spread being higher in the low speed limit than onthe high speed one), in agreement with measurements by HELIOS and ULYSSES spacecraft. The magnetic cycle of the Sun produces and modulates the solar wind, affecting space weather aroundEarth. Other than the available observational data, researchers rely on various kinds of numerical models tounderstand the solar magnetic cycle and related phenomena. The main objective is to forecast the futuresolar activity based on past observational data combined with numerical models.In this paper, we have presented a new 3D flux-transport dynamo model to understand the solar magneticcycle and how it affects the coronal topology, which is related with the generation of the solar wind. Wehave used a 3D kinematic dynamo model coupled with a 3D flux emergence model. In our model, thetoroidal flux at the base of the convection zone gets buoyantly transported to the outer surface to producetilted bipolar magnetic regions or sunspots. Later, the decay and dispersal of these sunspots generate aresultant poloidal field. The meridional flow then transports the surface magnetic flux towards the polesto reverse the polarities of the previous polar field. The newly generated poloidal field then gets shearedby the differential rotation to reproduce a toroidal field near the base of convection zone, completing themagnetic cycle. An additional nonlinearity introduced by putting lower and upper cutoffs on the emergingtoroidal flux results in the saturation of the magnetic energy, producing a self-sustained saturated dynamo.We note that this model is not intended to reproduce the full complexity of the solar magnetic field but onlythe evolution of the largest scales. Indeed, the model is for now kinematic (no back-reaction on the velocityfield), relies on a particular model (the Babcock-Leighton model) where the effect of convection is nottaken into account and finally dynamo action and emergence at small-scales are ignored.The combination of a large scale dynamo with localized magnetic flux emergence produces complextopological variations on the magnetic field of the corona throughout the whole activity cycle, that alsohave strong repercussions in the properties of the solar wind. We have analysed these effects altogetherby means of PFSS extrapolations of the surface field produced by the dynamo simulations, and of a newsolar wind model (MULTI-VP) constrained by the extrapolations. This novel approach lets us determinethe connections between the evolution of the dynamo field at the surface of the sun and that of the coronaand solar wind in a quick and precise way which can be easily repeated through the duration of oneor several solar cycles. The solar wind model provides a complete set of physical parameters such as
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Production of sunspots and their effects on the corona and solar wind the wind speed, density, temperature and magnetic field amplitude as well as derived quantities such asdynamical pressures and phase speeds, unlike more conventional semi-empirical scaling laws relating speedto magnetic geometry. The model also provides a sophisticated description of the wind thermodynamicsbetween the surface of the Sun and the high corona, unlike most global-scale solar wind models, that arecomputationally heavier and tend to rely on polytropic MHD or similar approximations. Our approachproduces more realistic solar wind properties (e.g., mass fluxes) than polytropic fluid models. MULTI-VP completely excludes the closed-field regions of the corona (e.g., streamers) from the computationaldomain, and reduce the physical description of the solar wind models to a collection of flows driven alongindividual flux-tubes. That way, we do not have to deal with cross-field interactions that would imposestrong constraints on the integration time-step that are unnecessary for the kind of steady-state solutionswe are looking for. But more importantly, there are positive trade-offs: this strategy lets us have a muchmore detailed description of the thermodynamics of the model than we would be able to in a full 3D MHDsetup, and the angular distribution of the flows is not limited by magnetic diffusion across the field (whichmakes, e.g., the transition between slow and fast wind flows excessively wide in global MHD models).These points are discussed more thoroughly in [32]. We have verified that the model reproduces manyof the main features of the coronal field at different moments of the cycle. We reported a slight tendencyto underestimate the open to closed magnetic flux ratio (and, correspondingly, overestimate the size ofstreamers). The general properties of the solar wind are generally very well reproduced, albeit with slowwind regions which are more extended than expected (which is consistent with the low open to closedmagnetic flux ratio).In conclusion, we have produced here a modeling chain which allows us to track the evolution of internaldynamo processes along with subsequent changes in the coronal and wind structures. This whole modelingchain lets us determine many synthetic observables such as white-light images of the corona (which canbe compared to coronograph imagery) and time-series of wind parameters at any position of the solaratmosphere (that can be compared with in-situ measurements by spacecraft). Some non solar-like featuresare present in the current setup used here, but these can be improved by modifying the parameters withinthe framework of our model. For example, there is a very significant overlap between two consecutivemagnetic cycles. Hence, this model is not fully capable of producing proper solar minima, which in turnaffects the solar wind solutions. Secondly, this model shows only few spots emerging at low latitudes,which can however be modified through another choice of meridional flow profile. The polar field strengthis too high compared to the Sun, which can be corrected by modifying the parameters of the buoyancyalgorithm (acting on the net emerging flux). In the present model, the rise velocity is constant for toroidalfields B lφ < B φ < B hφ . However, various studies suggest that the buoyancy should depend on the strengthof the toroidal field near the tachocline. First calculations however show that taking this dependencyinto account does not significantly affect the results presented here. These are the main areas where weneed to improve the current model to have a better match with solar observations. This work is thus onlya proof-of-concept that a model producing well-defined spots self-consistently coming from a buoyanttoroidal field does induce a cyclic reversal of the poloidal magnetic field and sustains dynamo action. Wemoreover show that this kind of model can be coupled with a solar wind model which shows the evolutionof the wind speed and density as a result of the dynamics of the dynamo-generated field.Future work will involve a more systematic comparison of the model predictions and real data inorder to improve the model’s methodology. Moreover, this model is a promising candidate for applyingdata assimilation techniques since it strongly relies on surface and solar wind features which areobserved. Moreover, thanks to our simplifying assumptions and numerical techniques, the whole chainis computationally cheap and thus very appealing for data assimilation where several realizations of the This is a provisional file, not the final typeset article umar et al. Production of sunspots and their effects on the corona and solar wind model are needed to minimize the mismatch with observations. Appying data assimilation would enable usto forecast the future solar magnetic activity and related phenomena. Finally, a modified version of thismodel could also be used to study the magnetic activity of other cool stars with an internal structure andvelocity profiles distinct from those of the Sun.
AUTHOR CONTRIBUTIONS
R. Kumar and L. Jouve developed the flux-transport model, performed the simulations, and studied the solarmagnetic cycle. R. F. Pinto and A. P. Rouillard carried out corona and solar wind studies. The manuscriptwriting was a combined effort of all the authors.
FUNDING
This work was supported by the Indo-French research grant 5004-1 from CEFIPRA, by the EC FP7 project
ACKNOWLEDGMENTS
Numerical simulations were performed on CALMIP supercomputing facility at Universit´e de Toulouse(Paul Sabatier), France, under computing projects and . REFERENCES [
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Production of sunspots and their effects on the corona and solar wind (a)(b)(c)(d)
Figure 7.
Magnetic field lines at the solar surface at the rise phase of the fifth cycle, at the reversal of thepolar field, at sunspot minimum, and immediately after the polarity reversal (as in the previous figures).The surface B r is rendered in gray scale (white for positive and black for negative polarities). The lines aremagnetic field lines extrapolated from the surface using PFSS, with source-surface radius at . (cid:12) . Greenand violet colours correspond to positive and negative B r . This is a provisional file, not the final typeset article umar et al. Production of sunspots and their effects on the corona and solar wind L a t i t u d e Longitude (a)(b)(c)(d) L a t i t u d e L a t i t u d e L a t i t u d e Figure 8.
Maps of the source regions of coronal holes at the surface. The colour scheme and the instantsrepresented are the same as in Fig. 7.
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Production of sunspots and their effects on the corona and solar wind
Time O p e n F l u x ( N o r m a li s e d ) Figure 9.
Open magnetic flux as function of time for the two half magnetic cycle. The black line showsthe total open flux measured at the source-surface (at . R (cid:12) ), while the blue and red lines represent highand low latitude open flux (beyond or within ± ◦ .) The high latitude open flux anti-correlates with thesunspot cycle phase, and dominates the total flux throughout most of the cycle. Longitude (c) L a t i t u d e L a t i t u d e L a t i t u d e L a t i t u d e (a) Velocity500-50
Figure 10.