Strong horizontal photospheric magnetic field in a surface dynamo simulation
aa r X i v : . [ a s t r o - ph ] J a n Astronomy & Astrophysics manuscript no. 8998 c (cid:13)
ESO 2018October 27, 2018
Letter to the Editor
Strong horizontal photospheric magnetic fieldin a surface dynamo simulation
M. Sch¨ussler and A. V¨ogler Max-Planck-Institut f¨ur Sonnensystemforschung, Max-Planck-Strasse 2, 37191 Katlenburg-Lindau, Germany Sterrekundig Instituut, Utrecht University, Postbus 80 000, 3508 TA Utrecht, The Netherlandse-mail:
[email protected], [email protected]
October 27, 2018
ABSTRACT
Context.
Observations with the
Hinode spectro-polarimeter have revealed strong horizontal internetwork magnetic fieldsin the quiet solar photosphere.
Aims.
We aim at interpreting the observations by means of results from numerical simulations.
Methods.
Radiative MHD simulations of dynamo action by near-surface convection are analyzed with respect to therelation between vertical and horizontal magnetic field components.
Results.
The dynamo-generated fields show a clear dominance of the horizontal field in the height range where the spec-tral lines used for the observations are formed. The ratio between the averaged horizontal and vertical field componentsis consistent with the values derived from the observations. This behavior results from the intermittent nature of thedynamo field with polarity mixing on small scales in the surface layers.
Conclusions.
Our results provide further evidence that local near-surface dynamo action contributes significantly to thesolar internetwork fields.
Key words.
Sun: magnetic fields - Sun: photosphere - MHD
1. Introduction
The ubiquitous existence of small-scale ‘internetwork’magnetic fields of mixed polarity in the so-calledquiet solar photosphere is strongly indicated by vari-ous observational diagnostics (e.g., Khomenko et al., 2003;Lites & Socas-Navarro, 2004; Trujillo Bueno et al., 2004,and further references therein). Recent high-resolutionspace-borne observations with the spectropolarimeter ofthe Solar Optical Telescope aboard the
Hinode satel-lite have considerably strengthened the case for inter-network fields and, furthermore, have revealed that themeasured internetwork flux is dominated by stronglyinclined, almost horizontal magnetic field (Lites et al.,2007; Orozco Su´arez et al., 2007b). Considerable amountsof highly time-dependent horizontal magnetic flux have alsobeen found in ground-based observations with lower spa-tial resolution (Harvey et al., 2007). The ubiquity of thesmall-scale, mixed-polarity internetwork field suggests a lo-cal origin of at least a significant part of the measuredflux. Recently, we have demonstrated by means of radia-tive magneto-convection simulations that local dynamo ac-tion by near-surface convective flows is a possible source forthe internetwork flux (V¨ogler & Sch¨ussler, 2007). Here weshow that the spatial structure of the dynamo-generatedfield provides a natural explanation for the observed domi-nance of the horizontal field component in the middle pho-tosphere.
2. Numerical model
We use the results of dynamo run C of V¨ogler & Sch¨ussler(2007) with 648 × ×
140 grid cells in a computationalbox with a physical size of 4 . × .
86 Mm in the horizon-tal and 1.4 Mm in the vertical direction, the latter rangingfrom about 900 km below to 500 km above the averagelevel of continuum optical depth unity at 630 nm wave-length ( τ = 1). The simulation has been run with the MURaM code (V¨ogler, 2003; V¨ogler et al., 2005). With amagnetic Reynolds number of about 2600, the simulationshows an exponential growth of a weak seed field with an e -folding time of about 10 minutes. The magnetic energysaturates at about 2.5% of the kinetic energy of the convec-tive flows, the maximum of the spectral energy distributionlying at horizontal spatial scales of a few hundred km, atwhich scales the field displays a distinctly mixed-polaritycharacter.
3. Relation of horizontal and vertical fieldcomponents
Figure 1 is based upon a snapshot from the saturated phaseof the dynamo run. The intensity image in the upper leftpanel shows the granulation pattern, which is almost undis-turbed by the presence of the magnetic field. The distri-bution of the vertical field component on the level sur-face τ = 10 − (upper middle panel) reveals the mixed-polarity nature of the dynamo-generated magnetic field,which preferentially resides in the intergranular downflowlanes. Note that the distribution of vertical field at this level M. Sch¨ussler and A. V¨ogler: Strong horizontal photospheric magnetic field
Fig. 1.
Snapshot from the saturated phase of the dynamo run C (V¨ogler & Sch¨ussler, 2007). The panels in the upper rowshow the continuum intensity at 630 nm wavelength (upper left) , a gray-scale image (‘magnetogram’, saturated at ±
10 G,black and white colour indicating the two polarities) of the (signed) vertical field component on the surface τ = 10 − ,roughly corresponding to the formation height of the spectral line used for the observations (upper middle) , and theunsigned horizontal field strength (upper right) , black colour in this panel representing very weak field and white colourindicating the saturation level of 10 G. The two panels in the lower row show gray-scale images of the logarithm of thehorizontal field strength (black colour indicating fields below 1 G) for vertical cuts through the simulation box along ahorizontal line at y = 1 . y = 3 Mm (right), respectively, in the upper panels. The level of τ = 1 isroughly at a height of 0 . τ = 10 − at 1 . τ = 10 − the latter (shown in theupper right panel) dominates in most places. The represen-tation on the two vertical cuts shown in the lower row ofFig. 1 illustrates that the horizontal field in the photosphere(i.e., above a height of about 0 . τ . The dashed curve gives theaverage unsigned vertical field while the dotted and dash-dotted curves represent the averages of the two horizontalfield components (unsigned).The averages of the three components indicate that thefield is not far from being statistically isotropic in the deeplayers below τ = 1. In contrast, the horizontal compo-nents of the magnetic field become increasingly dominantin the photosphere above. The driving of the dynamo bysmall-scale turbulent shear flows in and adjacent to the in-tergranular downflows is mainly restricted to the regionsbelow τ = 1 . , whereas, in the convectively stable photo-sphere above, these flows are much weaker and the rate ofwork against the Lorentz force drops steeply with height.Since inductive effects have smaller influence on the field . Sch¨ussler and A. V¨ogler: Strong horizontal photospheric magnetic field 3 Fig. 2.
Profiles of magnetic field strength averaged oversurfaces of constant τ : unsigned vertical field ( h| B z |i ,dashed), unsigned horizontal field components ( h| B x,y |i ,dotted and dash-dotted, respectively), and root-mean-square of the horizontal field strength ( h B x + B y i / , solid).in the layers above τ = 1, the decay of the field withheight is mainly determined by its spatial structure at thesurface (particularly by the energy spectrum as a functionof horizontal wavenumber). This results in a steep declineof the unsigned vertical field with height as opposite polar-ities on small scales are connected by shallow loops withtypical length scales of a few hundred km, correspondingto the horizontal scale for which the magnetic energy spec-trum at τ = 1 reaches its maximum. It becomes plausiblethat this configuration leads at the same time to a less steepdecline of the horizontal field, if one considers the simple ex-ample of an arcade-like magnetic field with concentric semi-circular field lines: for increasing height, more and morefield lines turn over horizontally, so that the horizontallyaveraged unsigned vertical field strength decreases fasterthan the averaged horizontal field; a simple calculation forthis case shows that h| B hor |i strongly exceeds h| B vert |i atheights of the order of the horizontal scale (footpoint sep-aration at the surface) of the arcade. Thus, the dominanceof horizontal fields in the photosphere is consistent withthe assumption of a simple loop topology with a preferredlength scale.The quantity that is actually relevant for a qualitativecomparison with the ‘apparent’ horizontal field strength de-rived by Lites et al. (2007) from measurements of the linearpolarization (transversal Zeeman effect, Stokes Q and U )is the root-mean-square of the horizontal magnetic field,i.e., B rms ≡ h B x + B y i / , since for not too strong fieldsStokes Q and U are proportional to the square of the hor-izontal field strength. B rms as a function of optical depthis shown as the solid line in Fig. 2. Owing to the inhomo-geneity of the horizontal field, this quantity is significantlylarger than h| B x |i and h| B y |i , even if we multiply any ofthem by a factor √ Fig. 3.
Ratio of the root-mean-square of the horizontal fieldcomponent (solid line in Fig. 2) to the averaged unsignedvertical field (dashed line in Fig. 2). In the optical depthinterval − < log( τ ) < − Hinode
SP), the ratio is roughlyin the range 4–6.izontal field. We obtain a number between 0.25 and 0.3in the range − < log τ < −
1, which is the relevantrange of line formation of the FeI lines at 630.15 nm and630.25 nm used by the
Hinode spectro-polarimeter (e.g.,Orozco Su´arez et al., 2007a). These values are consistentwith the estimate of the fill fraction obtained by Lites et al.(2007) and Orozco Su´arez et al. (2007b) by means of an in-version method.Let us now consider the ratio between the average ver-tical field and B rms as shown in Fig. 3. The ratio increasesstrongly with height, so that in the optical depth interval − < log τ < − B L app /B T app =55 G /
11 G = 5. However, as pointed out by these authors,the relation between B T app and the actual horizontal fieldstrength is far from trivial. Furthermore, effects of line-of-sight integration and spatial smearing by the instrumentcomplicate the relationship between the average fields inthe simulation and the field strengths derived from theobserved Stokes profiles. A direct quantitative comparisonwith the observations would have to proceed by means ofcalculating synthetic Stokes profiles, taking into accountthe point-spread function of the instruments. This is be-yond the scope of this Letter .Lites et al. (2007) have suggested that one possibilitycontributing to the imbalance of the average vertical andhorizontal fields could be a significantly larger horizon-tal scale of the horizontal field as compared to the verti-cal field. In fact, this is what we clearly find in our dy-namo simulation. Fig. 4 shows spectral magnetic energy asa function of horizontal wave number. The dashed curvegives the energy distribution for the vertical field, whilethe solid curve represents the spectral energy in the hor-izontal field (mean of the spectra for the two horizontal
M. Sch¨ussler and A. V¨ogler: Strong horizontal photospheric magnetic field
Fig. 4.
Magnetic energy spectra as a function of horizontalwave number, k , for the height range roughly correspondingto − < log τ < −
1. The dashed curve shows the energyspectrum based on the vertical field component, while thesolid curve gives the arithmetic mean of the spectra for thetwo horizontal field components. For low wavenumbers, theenergy in the horizontal field clearly dominates.field components). For this plot we have considered fieldsin the height range roughly corresponding to the opticaldepth interval − < log τ < −
1, which is relevant forthe formation of the iron lines used for the observations.The curves show that the field components are in equipar-tition at small scales (large wave numbers), but that thehorizontal field clearly dominates at wave numbers belowroughly 10 Mm − , corresponding to horizontal scales largerthan about 600 km.
4. Discussion
A direct comparison of the simulation results with the ob-servations is not possible because the individual values forthe averaged fields in the simulation are both (by about afactor three) smaller in the relevant height range than thevalues for the ‘apparent’ derived from the observation, thediscrepancy probably becoming even more severe when theactual spatial resolution of the observations is taken into ac-count. This is not particularly surprising since the magneticReynolds number of the simulation is still orders of mag-nitude smaller than the actual value in the correspondingsolar layers, so that the saturation level of our simulation isprobably considerably below the level to be expected for thereal Sun. In fact, a preliminary simulation with a roughlydoubled Reynolds number of about 5000 shows an increaseof the magnetic energy by a factor of about 1.7 with respectto the case shown here. Interestingly, it turns out that theoptical depth profiles of the averaged field strengths in thiscase can be very well approximated by just multiplying thecurves shown in Fig. 2 by √ .
7, meaning that the main dif-ference between the simulations reduces to a simple scalefactor for the field strength. Together with the fact thatthe observed internetwork fields are predominantly weak and their energy is significantly smaller than the kineticenergy of the convective motions, this suggests that thegeneral nature of the dynamo-generated field may not besignificantly different from the case shown here, apart froma higher overall amplitude. In particular, we expect thatthe ratio of the average horizontal and vertical field is notstrongly affected by the amplitude of the dynamo-generatedfield. Of course, these assertions need to be demonstratedby further simulations with higher Reynolds numbers.The clear dominance of the horizontal field in the midphotosphere seems to be a rather specific property of thestrongly intermittent field generated by near-surface tur-bulent dynamo action. Accordingly, the dynamo simula-tion of Abbett (2007, with a closed bottom boundaryand a local treatment of radiative transfer) also exhibitsstrong horizontal field in the photospheric layers. On theother hand, models with an imposed net vertical flux (e.g.,V¨ogler et al., 2005) or our recent simulations of the de-cay of a granulation-scale mixed-polarity field (at magneticReynolds numbers below the threshold of dynamo action)do not show this behavior; in these cases, the intricatesmall-scale mixing of polarities that is characteristic for thedynamo does not dominate the field structure. The rapiddecay with height of such a dynamo-generated field is alsoconsistent with the apparent lack of strong horizontal fieldin the chromosphere (Harvey et al., 2007).What are the alternatives to near-surface dynamo ac-tion? ‘Shredding’ of pre-existing magnetic flux (remnants ofbipolar magnetic regions) cannot explain the large amountof observed horizontal flux since the turbulent cascade doesnot lead to an accumulation of energy (and generation ofa spectral maximum) at small scales. On the other hand,such a behavior is typical for turbulent dynamo action. Fluxemergence from the deeper convection zone in the form ofgranule-sized small bipoles would have to proceed such ahigh rate in order to maintain the ubiquitous strong hor-izontal fields that it probably would not have gone un-detected in the past (see, however, Centeno et al., 2007).The sporadic appearance of horizontal internetwork fields(HIFs) described by Lites et al. (1996) and interpreted assmall-scale flux emergence events seems to be unsufficientto explain the ubiquitous horizontal field now found with
Hinode . On the other hand, flux recycling of an overall back-ground flux by granulation probably represents a signifi-cant source of horizontal field in network and plage regions.Emergence of extended horizontal field strands in granulesas observed by Ishikawa et al. (2007) is not seen in localdynamo simulations.In the real Sun, probably all three sources, i.e., dy-namo, shredded fields, and small-scale flux emergence fromdeeper layers, contribute to the internetwork flux in un-known amounts. In any case, the strong horizontal fieldsin the quiet photosphere inferred by the observations in-dicate that the source of these fields at the solar surfaceis a mixed-polarity field whose energy is mostly containedin those spatial scales where the dynamo-generated fluxresides. Therefore, the observational results obtained withthe
Hinode SP together with the analysis presented hereprovides strong indication that surface dynamo action rep-resents a significant source for the internetwork field in thesolar photosphere. . Sch¨ussler and A. V¨ogler: Strong horizontal photospheric magnetic field 5