Penumbral structure and outflows in simulated sunspots
PPenumbral structure and outflows in simulatedsunspots
M. Rempel ∗ , M. Sch ¨ussler , R.H. Cameron & M. Kn ¨olker High Altitude Observatory, NCAR, P.O. Box 3000, Boulder, Colorado 80307, USA Max-Planck-Institut f¨ur Sonnensystemforschung,Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany ∗ To whom correspondence should be addressed; E-mail: [email protected].
Sunspots are concentrations of magnetic field on the visible solar surface thatstrongly affect the convective energy transport in their interior and surround-ings. The filamentary outer regions (penumbrae) of sunspots show system-atic radial outward flows along channels of nearly horizontal magnetic field.These flows were discovered 100 years ago and are present in all fully devel-oped sunspots. Using a comprehensive numerical simulation of a sunspot pair,we show that penumbral structures with such outflows form when the averagemagnetic field inclination to the vertical exceeds about 45 degrees. The system-atic outflows are a component of the convective flows that provide the upwardenergy transport and result from anisotropy introduced by the presence of theinclined magnetic field.
Sunspots are dark patches on the visible solar surface that harbor strong magnetic fieldsup to 4000 G (
1, 2 ). Their central region, the umbra, is the darkest part with a brightness ofabout 20% of the ambient value and a largely vertically oriented magnetic field; the brighter,filamentary penumbra shows a more inclined field and a nearly horizontal plasma outflow ofseveral km · s − , the Evershed flow, named after its discoverer ( ). While a number of simplified(and partly conflicting) models have been suggested to explain the structure and outflows ofpenumbrae ( ), a comprehensive theoretical understanding of the basic mechanisms does notexist .Here we present ab-initio numerical simulations of complete sunspots embedded in a re-alistic solar convection zone and atmosphere, including all relevant physical processes: com-pressible magnetohydrodynamics, partial ionization, and radiative energy transport. Previous1 a r X i v : . [ a s t r o - ph . S R ] J u l ttempts to simulate penumbral structure in small slab-like sections of sunspots (
5, 6 ) resultedin rather narrow penumbral regions. The generally used periodic boundary conditions at thesides of the computational box tend to suppress the extended horizontal field structures as-sociated with sunspot penumbrae. Hence we have carried out a simulation of a pair of bigsunspots (diameter 35 Mm) of opposite magnetic polarity, thereby facilitating the developmentof strongly inclined field between the spots. Our numerical box had a horizontal extension of Mm × Mm and a depth of . Mm. The spatial grid resolution was km in the horizontaldirections and km in the vertical. The sunspots evolved for . hours during the simulation,which was sufficient to study the penumbral structure and dynamics; processes that evolve onlonger time scales such as moat flows were not fully developed in this simulation. However, thesurface evolution of magnetic field shows clear indications of bipolar magnetic features trans-ported away from the spots beyond the penumbra boundary. This is reminiscent to observationsof so-called ‘moving magnetic features’ ( ) (SOM, movie1.mpg: ’Magnetogram’ movie dis-playing the temporal evolution of B z on the visible solar surface, the gray scale is ranging from − . kG to +3 . kG). More detailed information about the physical model, the numerical code,and the simulation setup is provided in the SOM.The simulated penumbrae show the largest extension between the sunspots of opposite po-larity (Fig. 1A). The periodic horizontal boundary conditions provide three different distances: Mm (middle of box) and Mm in the x -direction between opposite polarity spots, and Mm in the y -direction between same polarity spots. The spots show a dark umbra with somebrighter umbral dots, preferentially in the weaker spot on the left. A deep magnetic structureunderlies the visible penumbra, particularly so between the sunspots (Fig. 1B). A movie cov-ering hour of temporal evolution of the properties displayed in Fig. 1 (SOM, movie2.mpg)shows the inward progression of filaments in the inner penumbra.The umbral regions have a brightness of . to . I , where I is the average quiet-Sunvalue, a Wilson depression of the visible surface by 550–600 km, and vertical field strength ( B z )up to 4 kG (Fig. 2). The quantities described here are averaged in space and time as describedin the caption of Fig. 2. The penumbrae have much weaker B z , horizontal fields ( B x ) withpeak values around 2 kG at the inner penumbral boundaries, and an average brightness of about . I . The penumbral region exhibit systematic outflows with average horizontal velocities ( v x )of up to 6 km · s − . The onset of these flows is closely related to the magnetic field inclination:where the average inclination with respect to the vertical exceeds 45 deg, there are systematicaverage outflows. With growing distance from the umbra, the outflow velocity increases and thefield becomes more inclined and is nearly horizontal in the outer penumbra. These propertiesare consistent with observational results (
1, 8 ).The simulated penumbra shows strong structuring in terms of elongated narrow filaments(Fig. 3). In the inner part the magnetic field shows strong variations of the inclination between40 deg and nearly 90 deg on scales of less than 200 km. Further out regions with stronglyinclined field dominate. The velocity structure is analogous: radial outflows are concentratedin highly inclined filaments and become stronger and azimuthally more extended in the outerpenumbra, where the field is almost uniformly horizontal. Vertical (up- and downward) flows2ccur in narrow filaments throughout the whole penumbral region.Analyzing the penumbral structure in vertical cuts we find that the outflows reach their peakvelocities (exceeding 10 km · s − ) near the visible surface (Fig. 4). This reflects the strong heightgradient of pressure and density in these layers: rising hot plasma turns over and the resultinghorizontal flow is guided outward from the spot by the strong and inclined magnetic field. Whilethe vertical field in the sunspot umbra only permits convection in the form of narrow columnarstructures ( ), the inclined field in the penumbra favours sheet-like upflows, which are radiallyextended and narrow in the azimuthal direction ( ). Together with the preferred weakening ofthe vertical field component due to flux expulsion by the expanding rising plasma, this explainsthe azimuthal structuring and large azimuthal variations of the field inclination in the innerpenumbra. The influence of horizontal flows on the field structure depends on the location inthe penumbra. In the inner penumbra they remain rather weak and have therefore only a limitedback reaction on the field structure. In the outer penumbra they become sufficiently strong tobend over field lines leading to more extended patches of horizontal field and flows. In additionto the strong localized outflows near the visible surface, there is a large-scale flow cell withplasma rising and diverging around the spot, as is evident by the general reddish color in therepresentation of the horizontal velocity in Fig. 4. Systematic inflows (comprising very littlemass flux) are apparent in the uppermost layers of the simulation box. Because these regionsare strongly affected by the upper (closed) boundary, it is not clear whether the inflows couldpossibly be related to the observed inverse Evershed effect in the chromosphere ( ).The central penumbral region between the spots has a mean bolometric brightness of I p =0 . I (averaged over y = ± . Mm from the midplane of the computational box and betweenthe central vertical dotted lines indicated in Fig. 2). The mean brightness of the upflow areas is . I p , while that of downflows areas is . I p . The corresponding values for undisturbed gran-ulation are . I and . I , respectively, implying they have similar properties. However, thepenumbral region shows an rms bolometric brightness contrast of 25.2%, which is substantiallylarger than the corresponding granulation value of 17.3%. Observations also imply a positivecorrelation between brightness and vertical flow direction ( ). This constitutes evidence for aconvective flow pattern that transports the energy flux emitted in the penumbra. Other studiesshow a correlation between intensity and line-of-sight velocities ( ), which for sunspots ob-served outside the center of the solar disk is dominated by the horizontal Evershed flow. This isconsistent with our findings, because in the penumbra the horizontal flow velocity is correlatedwith the vertical flow direction.Our detailed analysis (SOM) shows that the spatial scales of the flows providing the ma-jor part of the convective energy transport are similar for both undisturbed granulation andpenumbra. The primary difference is that there is no preferred horizontal direction for granula-tion, while the energy-transporting flows in the penumbra are distinctly asymmetric: convectivestructures are elongated in the radial direction of the sunspot. These properties were alreadyindicated in earlier simulations (
5, 6 ) and suggested as an explanation for the Evershed outflowin ( ). The simulation shown here confirms this suggestion and demonstrates the convectivenature of a fully developed penumbra. 3he horizontal asymmetry of the convective flows is also manifest in the correlation of . between the corresponding flow component ( v x ) and the brightness. We find that the rms ofthe outflowing velocity component ( v x ) in the penumbra is much larger than the transversecomponent ( v y ) (perpendicular to the filament direction), showing an asymmetry similar to thatfound by the scale analysis. The total rms velocity profile as a function of depth is very similar toits counterpart for undisturbed granulation, apart from a slightly higher peak value, confirmingthe physical similarity of convection in granulation and penumbra.The mass flux and energy flux show similar properties with respect to the length scalesand asymmetry (SOM), indicating that most of the outflowing material emerges, turns over anddescends within the penumbra. In the deeper layers, there is some contribution (of the order of10–20%) to both energy and mass flux by the large-scale flow cell surrounding the sunspots.The analysis of our simulations clearly indicates that granulation and penumbral flows aresimilar with regard to energy transport; the asymmetry between the horizontal directions andthe reduced overall energy flux reflect the constraints imposed on the convective motions by thepresence of a strong and inclined magnetic field. The development of systematic outflows is adirect consequence of the anisotropy and the similarities between granulation and penumbralflows strongly suggest that driving the Evershed flow does not require physical processes that gobeyond the combination of convection and anisotropy introduced by the magnetic field. Weakerlaterally overturning flows perpendicular to the main filament direction explain the apparenttwisting motions observed in some filaments (
14, 15 ) and lead to a weakening of the magneticfield in the flow channels through flux expulsion ( ).Although our simulations of large sunspots is realistic in terms of relevant physics, it doesnot faithfully reproduce all aspects of the morphology of observed penumbral filaments. Thepenumbral regions are considerably more extended than in previous local simulations, but theyare still somewhat subdued, probably owing to the proximity of the periodic boundaries. Thefilaments in the inner penumbrae appear to be too fragmented and short, dark lanes along brightfilaments ( ) form only occasionally, likely a consequence of the still limited spatial resolutionof the simulation. Finally, the initial condition of the magnetic field underlying the sunspotis quite arbitrary, owing to our ignorance of the subsurface structure of sunspots. Notwith-standing these limitations, the present simulations are consistent with observations of globalsunspot properties, penumbral structure, and systematic radial outflows. These and earlier sim-ulations (
5, 6, 9 ) suggest a unified physical explanation for umbral dots as well as inner andouter penumbrae in terms of magneto-convection in a magnetic field with varying inclination.Furthermore a consistent physical picture of all observational characteristics of sunspots andtheir surroundings is now emerging.
References and Notes
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Nature , 151 (2002).17. High-performance computing resources were provided by NCAR’s Computational and In-formation Systems Laboratory. The National Center for Atmospheric Research is sponsoredby the National Science Foundation. 5igure 1: Snapshot from the simulation. ( A ) Surface brightness map of the sunspot pair andthe surrounding convective pattern (granulation). ( B ) Color representation of the field strength(saturated at 8 kG) in a vertical cut through the midplane of the simulation box at y = 25 Mm.The vertical direction is stretched by a factor of 2. The white line indicates the height level ofthe visible surface (optical depth unity). 6igure 2: Horizontal profiles of various quantities at the visible surface, averaged over ± . Mm from the midplane in y and time-averaged over one hour. ( A ) Brightness (normal-ized to the average over non-magnetic regions, solid) and depression of the visible surface withrespect to non-magnetic regions (Wilson depression, dashed). ( B ) Strength of the vertical ( B z ,solid) and horizontal ( B x , dashed) magnetic field components. ( C ) Horizontal velocity ( v x ,solid) and magnetic field inclination with respect to the vertical (dashed). Vertical dotted linesindicate the regions with systematic penumbral outflows.7igure 3: Magnetic field and velocity structure at the visible surface for the sunspot on theright side of Fig. 1. ( A ) Vertical field component (saturated at ± . kG). ( B ) Inclination angleof the magnetic field with respect to the vertical direction (grey indicates | B | < G). ( C )Radial outflow velocity (saturated at ± km · s − , red indicates outflows). ( D ) Vertical velocity(saturated at ± km · s − , red indicates downflows). The vertical lines in A indicate the positionsof the cuts shown in Fig. 4. 8igure 4: Vertical cuts through the penumbra (indicated by the black lines in Fig. 3A). Thevertical direction is stretched by a factor of 2. Shown are the horizontal velocity component v x (left) and the field inclination (right). ( A, B ) Inner penumbra (right line in Fig. 3A). (
C, D )Outer penumbra (left line in Fig. 3A). The color representation of v x is saturated at ± km · s − .Black lines indicate the visible surface (optical depth unity).9 ppendix: Supporting online material (SOM)1 Simulation setup The simulation presented here has been carried out with the
MURaM
MHD code (
1, 2 ), withmodifications described in ( ). The physics, numerics and boundary conditions are similar toearlier runs described there, the primary difference here is the far larger domain size and theinitial magnetic field configuration.It is currently still out of reach to run an ab-initio simulation of the formation of an activeregion, primarily due to the large vertical extent of the simulation domain required for thispurpose. In this study we focus on a . Mm deep domain to study the near-surface structure of apair of opposite polarity sunspots. We start with a × × . Mm domain of thermally relaxedconvection, in which the τ = 1 surface is located about km beneath the top boundary. Themagnetic field is initialized as a pair of axisymmetric opposite-polarity sunspots, which arebased on the self-similar field configuration used by ( ). Each spot comprises about . · Mx magnetic flux, but their initial field strength is different ( kG and kG, respectively, at thebottom of the domain, dropping to about kG in the near photospheric layers). In the courseof the simulation, this leads to a pair of sunspots with a photospheric field strength of about . kG (spot on the left) and . kG (spot on the right). The separation of the two spots inthe middle of the domain is Mm, about Mm less than half of the horizontal extent in the x -direction. Owing to the periodic horizontal boundary condition this setup allows to study avariety of different combinations of field strength and inclination angles: In the x -direction wehave opposite polarity spots with . and . kG strength and separations of and Mm inbetween, in the y -direction the magnetic field is less inclined since periodicity imposes samepolarity spots in a distance of Mm. As a consequence we can study several realizations ofpenumbra in one simulation run and evaluate the robustness of our results.We ran the simulation for the first hour of simulated time with a rather low numerical gridresolution of × × km to get past initial transients. The second hour was performed at amedium resolution of × × km and then followed by another . hours with a resolutionof × × km (corresponding to × × grid cells). The results presented hereare based on snapshots near the end of the high-resolution run and partly on temporal averagesover the last hour. The total duration of the run of about . hours is still very short comparedto the typical lifetime of a sunspot, so that the umbral regions are not yet completely thermallyrelaxed . However, the dynamical time scales for the penumbral regions is much shorter andno significant change or trend of the properties discussed here has been observed over the . hours duration of the high resolution run. 10 Scales and anisotropy of energy and mass flux
The vertical energy flux is given by F z ( x, y, z ) = (cid:37)v z (cid:32) ε + p(cid:37) + 12 v (cid:33) + F Mz (1)Here ε denotes the specific internal energy and F Mz the vertical component of the Poynting flux.In the following discussion we ignore the Poynting flux, which does not exceed a few percentof the total energy flux.To quantify the scale dependence as well as possible anisotropies we consider here a dimen-sionless measure for the decorrelation of the energy flux when mass flux and specific enthalpyare smoothed independently over a certain length scale L . We define here dimensionless func-tions of the height z and smoothing length L that quantify the decorrelation of the energy fluxfor smoothing in the x -direction, P x , and in y -direction, P y : P x ( z, L ) = (cid:82) (cid:104) F z (cid:105) xL d x d y (cid:82) F z d x d yP y ( z, L ) = (cid:82) (cid:104) F z (cid:105) yL d x d y (cid:82) F z d x d y . (2)Here the quantity (cid:104) F z (cid:105) xL (and equivalent (cid:104) F z (cid:105) yL ) is given by (cid:104) F z (cid:105) xL ( x, y, z ) = (cid:104) (cid:37)v z (cid:105) xL (cid:104) ε + p(cid:37) + 12 v (cid:105) xL . (3)The brackets indicate smoothed quantities (to remove the contributions of scales smaller than L ), which are defined through a convolution using a Gaussian G L with the length scale L as fullwidth at half maximum. For example, a 1-dimensional smoothing in the x -direction is definedas (cid:104) f (cid:105) xL ( x, y, z ) = (cid:90) f ( x (cid:48) , y, z ) · G L ( x − x (cid:48) ) d x (cid:48) . (4)Smoothing vertical mass flux and the specific enthalpy guaranties a balanced mass flux ateach scale L , provided that the original mass flux was balanced within the domain. In generalthe vertical mass flux is not necessarily balanced within a subdomain, owing to the presence ofvertical oscillations and, possibly, of flows on scales larger than the subdomain. A meaningfuldetermination of the convective energy flux requires that these contributions must be eliminated.We achieve this by subtracting, at each height level, the density-weighted vertical mean velocity ¯ v z ( z ) = (cid:82) (cid:37)v z d x d y/ (cid:82) (cid:37) d x d y from the vertical velocity component before energy and massfluxes are computed. As a consequence, we include only the contribution from motions thatoverturn within the subdomain boundaries. 11 .2 Mass flux We also apply the same method to analyze the scale dependence and anisotropy of the massflux. To this end we smooth (cid:37)v z as described above and compute the quantities Q x and Q y inthe following way: Q x ( z, L ) = (cid:82) |(cid:104) (cid:37)v z (cid:105) xL | d x d y (cid:82) | (cid:37)v z | d x d yQ y ( z, L ) = (cid:82) |(cid:104) (cid:37)v z (cid:105) yL | d x d y (cid:82) | (cid:37)v z | d x d y (5)Since the mass flux is balanced, i.e. (cid:82) (cid:37)v z d x d y = 0 , we consider here the absolute value. Ouranalysis will therefore provide the typical scale on which most of the mass flux turns around. To compare the properties of the energy flux in a penumbral region and almost undisturbedgranulation we apply this procedure to two subdomains indicated in Fig. 5. Both regions haveabout × Mm horizontal extent (note that both boxes are contiguous owing to the periodicboundary condition). Fig. 6 presents the scale dependence and anisotropy of the energy fluxcomparing the granulation and penumbral region. Panel a) shows the geometric average of P x and P y for granulation. Different height levels are color-coded, with depth increasing fromblue toward red. The uppermost height level corresponds to the position at which the averagepressure scale height is km, the distance between the height levels is km. The totalrange covered in the vertical direction is about . Mm, excluding the lower most kmthat are affected by the boundary condition. With increasing smoothing length scale L (thuseliminating the contribution from small scales) the remaining convective energy flux decreasesand drops by a factor of about when L reaches a value of about km in the near photosphericlayers. For the deeper layers the corresponding curves are shifted toward larger values of L (approximately L ∼ H p ). The anisotropy, defined as P y /P x , (not shown here) is very closeto with fluctuations of up on larger scales. Panels b) and c) present the results fromapplying the same procedure to the penumbral region between the two spots as indicated bythe central white box in Fig. 5. While the overall shape of the curves remains the same, somedistinct differences occur for P x and P y . In the case of P x , the depth dependence is stronglyreduced and all curves almost coincide with a curve corresponding to that from about − Mmdepth in the case of granulation. On the other hand the depth dependence of P y is similar tothat of granulation, but overall the scales are reduced by a factor of about . Panel d) showsthe anisotropy defined through the ratio P y /P x . For scales of less than about km, thedeep layers remain close to isotropic, while the anisotropy reaches values of about . in thenear photospheric layers. According to our definition of the anisotropy values < indicatesmaller scales in the y as compared to the x -direction (i.e. the quantity (cid:82) (cid:104) F z (cid:105) yL d x d y fallsoff quicker than (cid:82) (cid:104) F z (cid:105) xL d x d y with increasing L ). For scales larger than km the deeper12ayers have anisotropies > , indicating a contribution from large scale flows more coherent inthe y direction. In the deep layers contributions from large scales in P y remain at about compared to about in undisturbed granulation. The difference of can be attributed tothe presence of the large scale Evershed/moat flow system in the penumbra region.On a qualitative level the behavior of the mass flux (Fig. 7) is not distinctly different fromthat of the energy flux. Overall the typical scale for the overturning of mass is larger thanthe typical scale for the energy transport in both granulation and penumbra. The degree ofanisotropy of the mass flux is reduced compared to the anisotropy of the energy flux. Theincrease of anisotropy on larger scales is more pronounced on all height levels. In the deeplayers the Evershed/moat flow contribution to the mass flux is about to (values of Q y remain around as compared to in the case of granulation).While the penumbral region shows distinct differences from granulation in terms of aniso-tropy in height levels extending several Mm downward, we find no significant change in therelative contribution of large and small scales in the overall energy transport. The presenceof large scale contributions is more prominent in the mass flux, but not substantially enlargedcompared to granulation.The similarities between energy and mass transport in granulation and penumbra are alsoevident from the height dependence of rms velocities (Fig. 8). The computation of the rmsvelocities in the penumbra is based on a smaller subdomain of half the size as indicated inFig. 5 to exclude umbral regions which would lower the overall rms velocity (for computingthe rms velocities a balanced mass flux within the subdomain is less important). We presentin panels a) and b) v rms (black) v x rms (red), v y rms (blue) and v z rms (green) for granulation andpenumbra, respectively; panels c) and d) display the rms velocities normalized by the totalrms. A comparison for the profile of v rms between granulation and penumbra does not reveala significant difference, except for slightly larger velocities in the photosphere and a steeperincrease toward the photosphere. In the case of granulation vertical motions contribute about to the kinetic energy, while both horizontal components contributing about . Thesecontributions are almost independent of depth except near the bottom where v z is preferredover horizontal motions due to the boundary condition. In the case of the penumbra the kineticenergy is dominated by the component in the direction of filaments ( x ) which contributes about , the contribution of vertical motions is reduced to about while horizontal motionsperpendicular to filaments contribute only about in the near photospheric layers.Overall we do not see a strong indication for a fundamental difference between energy andmass transport in granulation and penumbra, except for the anisotropy introduced by the pres-ence of strong horizontal magnetic field. Scales in the direction of the horizontal field becomeless dependent on the pressure scale height, while scales perpendicular to the preferred field di-rection remain strongly dependent on pressure scale height and are reduced. As a consequenceenergy and mass transport become more anisotropic toward the surface. The anisotropy leadsto a reduction of horizontal motions perpendicular to the field direction, while the componentparallel to the field is increased. The total rms velocity does not show a significant change,indicating that the kinetic energy of convective motions remains similar, it is just differently13istributed among the x , y and z direction. These similarities strongly indicate that driving theEvershed flow does not require physical processes that go beyond the combination of convectionand anisotropy introduced by the magnetic field.14igure 5: White boxes indicate the domains for which we compare the properties of energy andmass flux. The central box encloses most of the penumbral region between both spots, the boxnear the corners encloses a region of equal size with almost undisturbed granulation.15igure 6: Scale dependence and anisotropy of energy flux in solar granulation and penum-bral region. Panel a) shows undisturbed granulation for reference. Different height levels arecolor-coded ranging from the photosphere (dark blue) to the bottom of the domain (dark red),spanning a total of . Mm. The black dashed lines in panels b) and c) correspond to the darkblue (near photosphere) and dark red (near bottom of domain) curves in panel a). Panels b) andc) display the corresponding quantities for the penumbral region, panel d) the derived anisotropyof the energy flux with respect to the x and y direction.16igure 7: Same as Fig. 7 for the mass flux as defined by Eq. (5).17igure 8: Comparison of rms velocities in granulation and penumbra. Panels a) and b) show v rms (black) v x rms (red), v y rms (blue) and v z rms (green) for granulation and penumbra, respectively.Panels c) and d) present v x rms , v y rms and v z rms normalized by v rms .18 eferences and Notes
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