MAGNETIC PROPERTIES OF THE SOLAR INTERNETWORK

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1 The Astrophysical Journal, 611: , 2004 August 20 # The American Astronomical Society. All rights reserved. Printed in U.S.A. MAGNETIC PROPERTIES OF THE SOLAR INTERNETWORK H. Socas-Navarro High Altitude Observatory, National Center for Atmospheric Research, Mitchell Lane, Boulder, CO ; navarro@ucar.edu V. Martínez Pillet Instituto de Astrofísica de Canarias, Vía Láctea s/n, La Laguna E38200, Tenerife, Spain; vmp@ll.iac.es and B. W. Lites High Altitude Observatory, National Center for Atmospheric Research, Mitchell Lane, Boulder, CO ; lites@ucar.edu Receivved 2004 January 4; accepted 2004 May 4 ABSTRACT Advanced Stokes Polarimeter observations are used to study the weakest polarization signals observed in the quiet photosphere with flux densities in the range of Mx cm 2, which are found in internetwork regions. Our analysis allows us to reach an unprecedented spectropolarimetric sensitivity at the cost of sacrificing spatial resolution. We find evidence for intrinsically different fields in granules and lanes and characterize the average properties of the weakest observable flux concentrations. The magnetic signals observed suggest a strong coupling between magnetic fields and convective flows. Upflows bring up weak fields (equipartition or weaker) to the surface, with stronger upflows carrying larger amounts of flux. The circular polarization profiles observed in the granular regions display a very strongly asymmetric shape, which contrasts with the less asymmetric profiles observed in the downflowing regions. At downflowing locations with speeds of 0.5 km s 1, both weak and strong fields can be found. However, when the downflow speed increases (up to about 1 km s 1 ) both the mean flux and the intrinsic field strength show a tendency to increase. The asymmetry of the circular polarization profiles also shows a clear trend as a function of magnetic flux density. Low-flux regions display the negative area asymmetry one naturally expects for field strengths decreasing with height embedded in a downflowing environment. As we move to stronger flux density locations, the well-known positive area asymmetry develops and reaches even higher values than those typically found in network regions. These results may have important implications for our understanding of the coupling between magnetic fields and convective processes that pervade the solar photosphere. Subject headinggs: line: profiles Sun: atmosphere Sun: magnetic fields Sun: photosphere 1. INTRODUCTION The supergranular cell interiors are known to be populated by magnetic flux concentrations with a weak, mixed polarity pattern. They are commonly referred to as the internetwork (IN) flux, in contrast to the stronger network fluxes sited in the supergranular boundaries. The distinction between network and IN flux is not a clear cut one, as the two flux systems interact and can display locally similar properties. So-called quiet-sun regions always contain some amount of flux that can be categorized as network flux but also show regions that barely exhibit any magnetic flux signatures. In recent years, the characteristics, generation, and evolution of these IN fluxes have received much attention both observationally (Sánchez Almeida 2003; Sánchez Almeida et al. 2003; Khomenko et al. 2003; Lites 2002; Socas-Navarro & Sánchez Almeida 2002; De Pontieu 2002; Sigwarth 2001; Sánchez Almeida & Lites 2000; Lites et al. 1999; Lin & Rimmele 1999) and theoretically (through the surface dynamo vs. recycling debate by Cattaneo 1999; Stein & Nordlund 2002). The difficulties encountered in ascribing a given signal to either network or IN make the comparison between the results of these papers somewhat difficult. Differences in observed spectral lines (visible vs. infrared; see Socas-Navarro & Sánchez Almeida 1 The National Center for Atmospheric Research ( NCAR) is sponsored by the National Science Foundation ), spatial resolution, polarization sensitivity, and calibration techniques complicate even further a proper comparison. Advanced Stokes Polarimeter (ASP; Elmore et al. 1992) data have been used in several of the above works as a source for relatively high spatial resolution (1 00 ), good polarization sensitivity (10 3 times the continuum intensity) visible-light data (see Lites 2002 and references therein). However, the properties of the IN fields as observed in the IR region (Lin & Rimmele 1999; Khomenko et al. 2003) and the predictions from numerical simulations, showing that the generation of field concentrations occurs more often in the equipartition regime (500 G in the photosphere) or below, call for an increase in the sensitivity of our visible observations. In this paper, we make an effort to describe magnetic flux signatures at the very low end of the detectable signals in the visible range. By sacrificing the spatial resolution of data taken with the ASP and averaging independent pixels, we increase the final signal-tonoise ratios (S/Ns) of the resulting profiles to unprecedented levels. We present Stokes V (circular polarization) profiles of IN regions with a sensitivity in the range of (2 5) ; 10 5, which corresponds to S/Ns as large as 5 ; 10 4, 50 times the usual values achieved in individual pixels. In this way new unobserved IN signatures become available for further characterization in terms of intrinsic field strength, profile asymmetries, etc. One of the best-known ingredients of the IN regions is the salt-and-pepper fields with mean fluxes that are typically

2 1140 SOCAS-NAVARRO, MARTÍíNEZ PILLET, & LITES Vol times smaller than the network fluxes (see, e.g., Wang et al. 1995). Lites (2002) identifies these fields (with an apparent flux density of around 18 Mx cm 2 ) 2 with the granular component observed in the IR by Lin & Rimmele (1999) and found intrinsic field strengths in the range of G (see also Keller et al for visible observations and Khomenko et al for more recent IR data). According to Lites (2002) these fields occupy 32% of a given supergranular interior. But the rest of the supergranluar cell is not always devoid of fields. Lites (2002) also finds a diffuse, much weaker IN component occupying large (a few arcseconds wide) areas of the cell s interior. This weak component has apparent flux densities in the range of 5 Mx cm 2 and is approaching the noise limit of ASP data (1.5 2 Mx cm 2 ). The main objective of this paper is to further identify the properties of these very low flux regions by averaging the Stokes profiles of those pixels that show signals in some predefined ranges. We include profiles that appear to be only noise (photon limited in the ASP) but that, after the averaging, show a clear Stokes V profile of a very low amplitude. The Stokes V signals we aim to study are so weak that they can occur anywhere in the photosphere, not only in the intergranular lanes. We thus separate in the analysis granules and intergranular lanes. While it is well known that the magnetic field is swept to the lanes by convective patterns (see, e.g., Schüssler 2001), one expects that the continuous generation/recycling of flux brings it up to the surface near upflow regions. Upflow concentrations may then be expected to become magnetized at a given sensitivity level. Indeed, some evidence for this magnetized nature of granulation upflows has been found by Lites et al. (1996) in the horizontal internetwork features (HIFs) as signatures of linear polarization transient events and more recently by De Pontieu (2002), who finds the sudden appearance of longitudinal flux near upflow locations in large granules that sometimes evolve into exploding granules (Hirzberger et al. 1999) after the emergence event. It is important to realize that if all the IN flux that we see is brought up to the surface by such emergence events, upflow centers frequently must be magnetized, and the detection of this magnetic flux is only a question of sensitivity and/or spatial resolution. In x 2, we describe the observational data set and its calibration for intercomparison with other work. The flux distribution of the selected IN regions is analyzed in x 3. The spatial averaging of the individual ASP pixels is explained in x 4, where we display the resulting Stokes profiles. The velocities, asymmetries, intrinsic field strength, and polarity distribution of the averaged profiles in the different categories are discussed in x 5. Finally, x 6 provides the main conclusions of this work. 2. OBSERVATIONS The data set used in this paper was obtained on 1994 September 29 at 14:55 UT. This is probably the highest quality ASP map of the quiet Sun existing in the literature. For this reason it has been used frequently to investigate network and IN fields (Lites et al. 1996; Sánchez Almeida & Lites 2000; Socas-Navarro & Sánchez Almeida 2002; Lites 2002). The spectrograph slit was scanned over a very quiet region of 2 By apparent flux density, we refer to the product of the filling factor times the intrinsic field strength times the cosine of the inclination angle; see Lites (2002) for a description. width 57 00, producing a three-dimensional data cube. The four Stokes parameters were recorded using ASP s fast modulation of a dual beam with cameras running at 60 Hz to minimize seeing-induced cross talk among the various Stokes parameters. In ASP, two CCD cameras record the two orthogonally polarized modulated beams simultaneously. The optics and cameras were set up to cover the spectral region with a sampling of 12.8 m8. The spatial sampling both along and across the slit direction was of 0B38, with the spatial resolution being approximately 1 00, although this figure varies along the scanning direction because of time variations in the seeing conditions. The exposure time used for each slit position was approximately 3.5 s, resulting in a noise level (measured as the standard deviation of Stokes V in the continuum) of 5 ; 10 4 times the continuum intensity. Figure 1 shows the continuum map (left), as well as a magnetogram generated by integrating the Stokes V signal over the blue lobe of the line (right). The (unsigned) magnetic flux density displayed in the image has been obtained using the so-called magnetograph formula. It is used in this paper as a reference for the amount of circular polarization signal and not as the actual solar flux (see, e.g., Landi Degl Innocenti 1992 for references and notation): V(k) gk B cos di(k) dk ; ð1þ where is the magnetic filling factor. This equation may be rearranged to express an approximate longitudinal flux density as a function of the measured polarization over a certain bandwidth defined by a spectral profile p(k): R 1 0 V(k)p(k)dk B jj C R 1 0 ½dI(k)=dkŠp(k)dk ; ð2þ where C is a constant [C ¼ 1=(4:67 ; k 2 g)]). The resulting B jj provides a measure of the amount of circular polarization in magnetic field units. With this calibration and using for p(k) a 200 m8 wide step function, the ASP noise is 1.8Mxcm 2. The magnetogram in Figure 1 (right) has been saturated at 30 G in order to display the weakest IN signals. The solid line represents the boundaries of the IN, defined manually by trying to keep the network elements outside. The criterion to define the IN is somewhat arbitrary because it is not always clear whether some particular flux concentrations are branching out of the network or just coincidentally aligned in the IN. The figure shows small magnetic knots (1 00 or 2 00 across) with polarization values larger than 20 G that seem to be arranged on mesogranular scales. Unfortunately, it is not possible to make any rigorous analysis of such scales because of the small field of view of our data set. In any case, the subjectivity in our definition of the IN does not significantly affect the results presented in this paper, although there is a possibility that some of the small magnetic elements mentioned above may belong to the network. 3. FLUX DISTRIBUTION Before analyzing the detailed spectral profiles it is worthwhile to consider the distribution of magnetic flux in the entire region (including the network). If we assume that the value given by equation (2) is a reasonable approximation to the flux

3 No. 2, 2004 LOW-FLUX SIGNALS IN INTERNETWORK REGIONS 1141 Fig. 1. Left: Continuum intensity in the observed region. Right: Unsigned degree of polarization measured in Gauss, saturated at 30 G to show network and IN magnetic structures. The contours have been defined by hand to enclose IN regions. density (at least in a statistical sense), then we can plot a histogram with the relative frequency of the various flux values. This is shown in Figure 2 (left). Note that the histogram is very well fitted by a combination of two exponential functions. We stress that the quantity plotted in Figure 2 is simply the flux density and not the intrinsic field strength, which, in general, cannot be retrieved without an appropriate inversion procedure. Nevertheless, it is possible to discriminate between strong and weak fields by using the two spectral lines in our observations. If the field is weak, then equation (1) is a good approximation and the flux densities obtained for both lines will be identical. However, since the line at has a larger Landé factor, it abandons the weak-field regime before the line does. As the field strength increases and the Zeeman splitting becomes larger than the Doppler width, the higher Landé factor results in lower flux values. If we construct a similar histogram but use only pixels that harbor weak fields (i.e., those in which the flux density obtained from both lines differs by less than 5 Mx cm 2 ), the distribution obtained is compatible with one of the exponential curves (see Fig. 2, right). Figure 2 suggests that the weak and strong components of the field obey different distribution curves. This might indicate two different physical mechanisms operating to produce these fields. However, the analysis presented in this section has important limitations that prevent us from making any definite claims. Perhaps the most severe limitation is the inadequacy of equation (1) to estimate the flux, especially for strong fields or in the presence of mixed polarities (which are detected more frequently in pixels with weaker flux; see Sánchez Almeida & Lites 2000; Socas-Navarro & Sánchez Almeida 2002). In any case, these distributions may provide important constraints for future numerical simulations. Instead of comparing these histograms directly with the flux distribution obtained from the simulations, it would be more interesting to synthesize the Fig. 2. Left: Histogram of magnetic flux density, as given by the magnetogram approximation. The vertical dashed line marks the noise level. The dotted lines represent two exponential curves whose addition (dashed curve) fits the distribution. Right: Same plot but using only pixels that meet the weak-field criterion (see text).

4 1142 SOCAS-NAVARRO, MARTÍíNEZ PILLET, & LITES Vol. 611 Fig. 3. Identification of granules (left) and intergranular lanes (right) in the IN superposed on the continuum image. corresponding spectral profiles to simulate observations and apply equation (2) to these profiles. 4. PROCEDURE The information on the magnetic field, encoded in the polarization profiles of the spectral lines, is buried deep within the noise in most of the IN pixels. We performed spatial averages of the spectral profiles over the entire IN. In doing so we have lost any information on the distribution of the magnetic structures over the field of view, but in return we are able to analyze average profiles from regions that do not exhibit polarization above our detection threshold. The averages are carried out separately for granules and lanes, the discrimination being based on their continuum brightness. Those pixels brighter than their surroundings (the average of an 11 ; 11 box surrounding the point under consideration) by at least the rms contrast of the granulation (between 3% and 4%) are taken to be granules. A similar criterion is used to identify intergranular lanes (points that are at least 3% darker than their surroundings). Visual inspection of the maps confirms that the selection of granules and lanes is correct, although an important fraction of points that could have been used in the analysis has been left out (see Fig. 3). By definition, our thresholding picks up only 35% of the IN pixels (the IN accounts for 43% of the map), with 18% being classified as granules and 17% as lanes. Furthermore, the profiles are classified according to the amount of polarization that they exhibit and then averaged separately. This allows us to analyze the dependence of the magnetic properties on the flux density. Three different ranges are considered, comprising signals from 1.5 to 3 Mx cm 2 (hereafter referred to as L1 signals), from 3 to 10 Mx cm 2 (L2 signals), and finally from 10 to 50 Mx cm 2 (L3 signals). L1 signals are well within the noise level and constitute the background of Figure 1 (right). L2 signals are at or slightly above the noise in the magnetogram and exhibit a diffuse appearance over the entire IN. L3 signals, on the other hand, are found in the IN as small localized flux concentrations. Some of the pixels in the L3 range (above 15 Mx cm 2 approximately) start to show recognizable profiles and were inverted by Socas-Navarro & Sánchez Almeida (2002). The spectral profiles obtained with this averaging procedure are plotted in Figure 4. Table 1 gives the percentage occurrence of each class among all the averaged pixels. The absolute number of lanes and granular pixels included in the average is given in the last row. The polarity of the field needs to be taken into account before averaging. Otherwise, opposite-polarity profiles would cancel out and leave no signal or only a small residual. We use the convention that positive-polarity profiles have righthanded circular polarization in their red lobe. If a given pixel exhibits negative polarity, the sign of Stokes V is reversed before averaging. Determining the polarity is not a trivial task when we are dealing with weak signals in which the polarization profiles are close to (or even below) the noise level. In order to make the discrimination as reliable as possible we take the average of 30 points to the red and blue of line center and compare their sign. In order to verify the reliability of our polarity classification, we performed a test using the solid-line profile of Figure 4d as a reference. This reference profile is scaled down to a certain amplitude, and then noise is added. For each amplitude, we consider 10 3 different realizations of the noise and apply our classification criterion. Table 2 presents a summary of the results. Note that, even for amplitudes as low as 5 ; 10 4, the probability for a successful classification is above 95%. The results presented in this paper are based on the assumption that the average profile over a certain region carries information on typical properties of that region. In other words, we assume that the average profile that we obtained is similar to the profile that would be observed at a hypothetical spatial location with typical magnetic and thermodynamical properties. Unfortunately, it is not possible to test the general validity of this approximation because the individual profiles that we are interested in are masked by the noise. However, it is possible to verify this assumption for the particular case of the stronger signals. Figure 4d shows the average profile of signals between 30 and 60 Mx cm 2 (solid line). Overplotted in that

5 No. 2, 2004 LOW-FLUX SIGNALS IN INTERNETWORK REGIONS 1143 Fig. 4. (a c) Average L1, L2, and L3 profiles, respectively. The solid (dashed) line represents the average lane (granule). (d) Average profile for signals between 30 and 60 Mx cm 2 (solid line) along with the profile from a particular single point that exhibits the average signal in that range (dashed line). figure is the individual profile observed at a particular spatial location that exhibits a degree of polarization of 41 Mx cm 2 (dashed line), which is the mean signal in the range. The good agreement between both profiles proves that our assumption is valid at least for the strong signals. In the remainder of this paper we extrapolate this result to the entire region, which implies that the magnetic properties inferred from our average profiles are representative of the averaged areas. This assumption, which is probably fairly reasonable, may be tested in the future with better quiet-sun observations. 5. RESULTS 5.1. Zero-CrossinggWavvelenggths Perhaps the most obvious difference between granule and lane profiles is a global wavelength shift at all signal levels, which indicates that the field is moving with the plasma following the convective granular motions. Table 3 shows the difference between the respective zero-crossing wavelengths TABLE 1 Percentage of Pixels in Each Signal Class Signal Range Granules Lanes L1 (%) L2 (%) L3 (%) All (number) and the Stokes V network (from the same map) zero-crossing wavelength. For strong fluxes, this reference is known to be very close to an absolute zero of velocity, or at most 200 m s 1 redshifted (see Martínez Pillet et al. 1997). The asymmetries of the profiles indicate that there are gradients of velocities along the line of sight, making the interpretation of these velocities not straightforward. While keeping this in mind, it is interesting to see how stronger signals correspond to stronger downflows for the lanes or more intense upflows for the granules. One concludes that whenever the convective flows are stronger (up or downward directed) the corresponding amount of flux carried or concentrated by them is larger. Interestingly enough, a similar trend has been found in the IR by Khomenko et al. (2003), who find higher intrinsic field strengths (in the range of G) for stronger TABLE 2 Probability of Correct Polarity Determination from a Sample of 10 3 Realizations for Each Amplitude Profile Amplitude Realizations above 1.5 Mx cm 2 (%) Success 5 ; ; ; ;

6 1144 SOCAS-NAVARRO, MARTÍíNEZ PILLET, & LITES Vol. 611 TABLE 3 Velocity from Stokes V Zero-Crossing Points with Respect to Mean Network Profile Signal Range (km s 1 ) Lanes (km s 1 ) downflows. For the upflows their conclusions are less indicative, however. The line at systematically exhibits larger velocities than that at (in absolute value) except for the L1 signals,wheretheyaresimilar.thecaseforthel3signalsis particularly clear. This line is weaker (has a lower opacity) than the line and therefore probes lower photospheric layers, where the convective motions are stronger. This result indicates that the magnetic field is moving down with the plasma Asymmetries The asymmetries exhibited by granules and lanes are qualitatively and quantitatively distinct. We may conclude from this fact that the structure of both the velocity and magnetic fields differs significantly between the two. We follow Martínez Pillet et al. (1997) for the definitions of Stokes V amplitude and area asymmetry: a ¼ a b a r ; a b þ a r P V A ¼ s P ; jvj (km s 1 ) Granules (km s 1 ) L L L where a b and a r are the unsigned extrema of the blue and red lobes of Stokes V, respectively, and s is the sign of the blue lobe. When these formulae are applied to our average profiles, we obtain the values in Tables 4 and 5. In the last row of each table, we provide the values obtained from the mean network profile on this map as a reference. The area and amplitude asymmetries in the plages and network of the Fe i lines have been studied extensively in the past using ASP data (Martínez Pillet et al. 1997; Sigwarth 2001). Amplitude asymmetries are always larger (about a factor of 2) thanareaasymmetries, asisthecaseforotherfe i spectral lines (see Solanki 1993 for a review). Traditionally, the area asymmetry has been explained by the canopy effect (Grossmann- Doerth et al. 1988). The area asymmetry arises in this model ð3þ ð4þ from those lines of sight that first traverse the flared, magnetized canopy of a flux tube, then traverse into its unmagnetized surroundings (usually an intergranular lane). The combination of downflow velocity in the lower, unmagnetized part, and a magnetic field at rest in the upper part of the atmosphere is able to reproduce Stokes V profiles with the observed area asymmetry. However, the amplitude asymmetry produced by this effect is of the same order of the area asymmetry, not a factor of 2 larger. As the observed amplitude asymmetries are double that of the area asymmetry, additional mechanisms, such as flux tube waves (see Solanki 1993), have been proposed to also reproduce the observed amplitude asymmetry. The MISMA scenario studied by Sánchez Almeida (1997) satisfactorily fits standard network profiles by using a suitable combination of correlations between magnetic field and velocities with fluctuations at length scales smaller than the photon mean free path. Bellot Rubio et al. (2000) used a thin (but optically thick) flux tube model to reproduce plage profiles of the same Fe i lines studied here. In this scenario, the inner core of the tube (excluding the canopy) produces a negative area asymmetry and positive amplitude asymmetry. Negative area asymmetries of the flux tube core are a natural consequence of the decreasing field strength and the downflow inside the tube used in this work. As expected, the surrounding canopy part of the tube produced large, positive contributions to both asymmetries. The smaller area asymmetry resulting is, then, a direct consequence of the contribution from the flux tube cores that provide a negative term. No flux tube waves are needed here to explain the excess of amplitude asymmetry (which should rather be thought of as a decreased area asymmetry). An important point to bear in mind is that, according to Bellot Rubio et al. (2000), the central core of the flux tube, with a downflow that increases with depth and a field strength that decreases with height, produces a negative area asymmetry contribution (as already predictedbysánchez Almeida et al. 1988). This picture provides a very appealing explanation for the excess amplitude asymmetry. However, it is important to point out that the assumption of a downflow inside the magnetic component in this model may not be justified. In particular, recent observations obtained with adaptive optics (Rimmele 2004; see also Sankarasubramanian & Rimmele 2003) suggest that the downflow inside the magnetic component (near the diffraction limit of the observations) is at most in the range of m s 1, whereas in the Bellot Rubio et al. (2000) model a value closer to 1000 m s 1 would be needed. A definitive resolution of this issue awaits full Stokes spectropolarimetry at an angular resolution comparable to that of Rimmele (2004). The asymmetries shown in Tables 4 and 5 can be interpreted in the light of the above discussion. We first concentrate on the case of intergranular lanes. The first remarkable point is that L1 signals show a clear negative area asymmetry. As pointed out earlier, this is the natural sign of the asymmetry for a flux TABLE 4 Stokes V Area Asymmetry Lanes Granules Signal Range A 6301 A 6302 A 6301 A 6302 L L L Network

7 No. 2, 2004 LOW-FLUX SIGNALS IN INTERNETWORK REGIONS 1145 TABLE 5 Stokes V Amplitude asymmetry Lanes Granules Signal Range a 6301 a 6302 a 6301 a 6302 L L L Network concentration embedded in a downflow and with the field strength decreasing with height. We expect that these low-flux signals are coupled to the convective motions and participate in the general downflow in intergranular lanes. Similarly, we also expect a quite natural decrease of the field strength with height due to the decreasing gas pressure in this direction. Thus, the negative area asymmetry becomes a rather natural outcome under these circumstances. It indicates that one has to turn, indeed, to these very low flux levels to find the negative asymmetries that would have been expected to occur more commonly. A transition to the observed positive area asymmetries occurs somewhere at L2 flux levels. For L3 signals it shows a clear positive value that is even larger than those found for the network. Within the canopy scenario (unmagnetized downflow in the bottom layers and a magnetized component at rest in the upper layers) for the generation of positive area asymmetries, one would argue that canopies are created at fluxes above L2. At weaker signals, a classical flux tube picture would not be needed. There a field strength decreasing with height and cospatial with a downflowing medium would easily generate the observed values. The amplitude asymmetries run from symmetric profiles in L1 flux levels to the usual positive values that also become larger than those observed in the network. Although we do not provide an explanation for the origin of these amplitude asymmetries, the correlation trend with increasing flux levels is very interesting. One could ascribe the amount of amplitude asymmetry to the intensity of the local downflows. Note in Table 3 that the downflows are stronger at higher flux levels (L3 signals) than for low-flux regions (L1 signals). This increasing downflow speed would naturally produce larger asymmetries (areas and amplitude) for any of the proposed scenarios mentioned above using velocity and magnetic field gradients. The reduction of the asymmetries when going to network flux levels also fits within this explanation. In the network, we already expect a reduction of the convective strength by the magnetic field itself and thus gentler flows as compared to normal granulation. We now turn to the case of granules. Amplitude asymmetries are always positive at all flux levels and increase for higher fluxes, as in the case of intergranular lanes, but reach values as large as 40%. Area asymmetries start at slightly negative values, of a few percent only, and become increasingly positive for L2 and L3 signal levels. Area asymmetries reach levels as large as 50%. We thus find a predominance of positive asymmetries in granular profiles that increase with flux levels and reach values larger than for intergranular profiles. The extreme asymmetry of the granular L3 profile is clearly displayed in Figure 5. This predominance of positive asymmetries can be understood as follows. In the same way we mentioned that the natural asymmetry for a field that decreases with height in a downflow is negative (and that is almost never found except for the L1 lane case studied here), a field that decreases with height in an upflow will have a positive asymmetry. Changing the sign of the velocity gradient simply flips the sign of the asymmetry accordingly (Landi Degl Innocenti & Landi Degl Innocenti 1981). One should see these granular fields 3 as the natural consequence of flux being brought up to the surface by convective cells and as observed by Lites et al. (1996; for the HIFs case) or by De Pontieu (2002). It is interesting to mention that, in the observations analyzed by De Pontieu (2002), flux emergence occurs near divergence centers where large granules develop and, sometimes, explode giving rise to a downflow in their centers. In this later case, one would have a downflow in the center of the granule with a field in the upper part of the atmosphere. This is exactly the same configuration as in the canopy scenario explained above, and it again gives rise to positive asymmetries. Thus, if flux emergence has a natural tendency to display positive asymmetries, the subsequent development into an exploding granule will reinforce this tendency. These two effects can explain the observed predominance of the positive asymmetries in the granular profiles Field Strenggths Socas-Navarro & Sánchez Almeida (2002) used the ratio of the Stokes Vamplitudes at and to discriminate 3 The term granular fields is used in this work to refer to magnetic fields embedded within the granules. The same denomination was used by Lin & Rimmele (1999) to designate the IN fields found in their observations because the spatial scales of these fields are comparable to those of the granulation. However, their fields are located in the downflowing dark lanes. Fig. 5. Average Stokes V profiles in granules (same as displayed in Fig. 4 with dashed lines) for the L1, L2, and L3 signals (in increasing amplitude). Note the extreme asymmetry of the L3 granular signal.

8 1146 SOCAS-NAVARRO, MARTÍíNEZ PILLET, & LITES Vol. 611 TABLE 6 Ratio r of Stokes V Amplitudes Signal Range r lanes r granules L L L Network TABLE 7 Percentage of Pixels with the Dominant Polarity Signal Range Lanes (%) Granules (%) L L L between weak and strong fields. The argument is similar to that discussed in x 3, regarding the validity of equation (1) in the weak-field regime. If the field is weak, 4 then r ¼ a(v 6301 ) a(v 6302 ) ¼ g6301 ea g 6302 ea a(di 6301 =dk) a(di 6301 =dk) ; where a( f ) denotes the amplitude of a given spectral profile f. The right-hand side of equation (5) is close to 0.65 in the quiet Sun. This is then the value of r that one would expect in the presence of weak fields. Network profiles, with kg fields, show a saturation of the amplitudes of the Fe i and lines where both lines have the same amplitude (see also Fig. 4d). Table 6 lists the values of r (along with the expected errors) calculated from our average IN profiles. The bottom row shows the ratio for the network. One expects that intergranular lane signals with kg fields will display similar amplitudes for both Fe i lines. Granular profiles show amplitude ratios very close to the expected value for weak fields at all signal levels. Note that the expected ratio for weak fields of 0.65 is model dependent and the exact value in a granular atmosphere will be slightly different from the value in an intergranular lane. The ratios in Table 6 clearly show that the field lines advected by the granulation are in the weak-field regime. The weak-field regime applies in a range between 0 and somewhere near 500 G (which happens to be the kinetic equipartition value). Thus, the granular fields encountered in this work have an upper limit for their intrinsic field strength of 500 G. The signals found in the intergranular lanes behave in a different way. L1 signals are in the weak-field regime, so they have an upper limit for their field strengths similar to that of granular signals. But as the amount flux observed within the intergranular line increases, there is a tendency for the intrinsic fields strength to increase. L2 and L3 signals shows ratios above the weak-field limit, so the intrinsic fields there must be necessarily stronger. Note that the ratios for L2 and L3 intergranular fluxes are also far from the saturation level shown in network profiles (which correspond to values around 1.5 kg). Again the comparison with the strength of the local downflows becomes relevant. The stronger the downflow, the higher the amplitude ratio in Table 6, indicating stronger intrinsic fields Polarity Consider now the polarity of the field, as defined in x 4. The percentage of points that exhibit the dominant polarity in the region are listed in Table 7. Both polarities occur with 4 Note that, as pointed out by Socas-Navarro & Sánchez Almeida (2002), if the field is weak, then it does not alter significantly the thermodynamical conditions of the plasma, which is also a necessary condition for the weakfield approximation to be applicable. ð5þ approximately the same frequency, except in lanes with L3 signals, where it is reminiscent of the network excess (Lites 2002). 5 The polarity distribution of the weaker signals (L1 and L2) is compatible with a local dynamo process, which would not favor any particular polarity. The strongest (L3) IN signals, on the other hand, may have a contribution from other sources, such as dissipating active regions or network elements. Lites (2002) concluded that a small imbalance on the order of 20% was present in the region when analyzing signals in the L2 and L3 regimes (over the whole map, defining the IN boundary as signals larger than 40 Mx cm 2 ). But in this work, IN areas in almost perfect balance of polarities were also found. 6. DISCUSSION AND CONCLUSIONS It is important to state clearly what signals have been analyzed in this paper and the main differences with previous works that have used the same data set. First of all, we have manually selected IN regions that explicitly avoid network concentrations. Only the outlined areas in Figure 1 are considered for our analysis. Within these regions a criterion that classifies the individual pixels in granules and lanes has been applied. This eliminated 65% of the data that have an intensity contrast intermediate between the granulation and lane definitions. A total of 3568 pixels are left for the analysis presented in this paper. A classification in terms of the signal level within each pixel is made for averaging within three polarization intervals. This also included an interval for pixels displaying no apparent signal above the noise. To minimize the cancellation between very weak Stokes V signals of opposite polarity, a polarity assessment is derived from the average over 30 spectral points on either side of line center. In this way, spatially averaged Stokes V profiles for the ranges L1, L2, and L3 are obtained separately for bright granular regions and dark intergranular lanes. These profiles have S/Ns in the 10 4 regime and represent the weakest signals observed so far in the visible range. Sánchez Almeida & Lites (2000) and Lites (2002) have extensively studied this data set in order to understand the nature of quiet-sun areas. In the case of Sánchez Almeida & Lites (2000), Stokes inversions were performed, and the S/N of the input data needed to be above some threshold. In their case they analyzed profiles of type L3. Lites (2002) developed a calibration procedure to include even smaller signals than the former work. It is clear that the weak IN of Lites (2002), with signals in the range Mx cm 2, corresponds roughly to our L2 profiles. In that work only the spatial distribution and the mean flux levels were studied and not the detailed shapes of the profiles. The L1 signals analyzed here have never been studied in any previous work. L1 signals are buried in the noise of the individual pixels, and only 5 The analysis of Lites (2002) considers the entire map, including the network, and obtains a larger polarity imbalance of 20%.

9 No. 2, 2004 LOW-FLUX SIGNALS IN INTERNETWORK REGIONS 1147 through our spatial averaging have we been able to bring them to detectable levels. Note that this does not imply that we are able to overcome all the negative effects of a limited spatial resolution in the averaged profiles. Our resolution element of 1 00 has already added all signals contained in this area, and they might not contribute to our average profiles. If, for example, two opposite-polarity regions of similar properties are included in our resolution element, the complete cancellation of their signals cannot be recovered by the procedure employed here. Only with higher spatial resolution can this be improved. Thus, a mixed-polarity field of any nature that is structured at scales of 0B5 or below will remain invisible to this analysis. A mixedpolarity field with coherence regions larger than 1 00 and that was below the noise levels of the original data will, however, show up in the final profiles by our averaging procedures. The average profiles obtained in this work clearly indicate a strong coupling with the local convective processes. L1 signals in granules and lanes are weak fields, in the equipartition regime or below, that are carried by the convective cells. Intergranular L1 signals show the negative area asymmetry that one would expect for a field strength decreasing with height and a local downflow within the magnetic field. This clearly contrasts with the higher flux signals that show a positive area asymmetry, more reminiscent of the network profiles. If the explanation for these positive values is the existence of a canopy (Grossmann-Doerth et al. 1988), the evident conclusion is that a canopy topology only forms at fluxes L2 or above. The stronger downflows show the larger fluxes and the stronger intrinsic field strengths (as dictated by the amplitude ratio of Table 6). This correlation has also been found for IR profiles by Khomenko et al. (2003) with intrinsic field strengths in the range of G (cf. their Fig. 7) that would be compatible with the fields we expect for L1 and L2 signals. Granular fields are always weak, and the stronger the upflow, the more flux it carries. This component represents the emergence of flux at granular scales (De Pontieu 2002), and that occasionally could show horizontal field components giving rise to the HIFs (Lites et al. 1996). The asymmetry shown by these profiles is in accordance with a field strength decreasing with height. The fields brought up by the upflows can feed nearby downflows, where they eventually increase in strength, more likely at those places with a stronger downflow. On average, the IN fluxes found in this work have a polarity balance, but the nearby network fields do not (cf. Lites 2002 or our L3 signals). The whole picture is very appealing in the sense that it seems a clear expression of the convective fields similar to those found in numerical simulations (Cattaneo 1999; Stein & Nordlund 2002). Further analysis of the profiles found here, including their inversions, is left for future work. It is important to point out that the total unsigned flux we find in granular fields is only slightly smaller than the flux we find in the lanes. For example, L1 signals of lanes and granules are of similar amplitude (by definition) and occur in similar numbers (Table 1). Only for stronger signals does one find higher fluxes in the lanes than in the granulation. One could argue that the granular signals represent scattered light from the strong network and IN fields located in intergranular lanes. Although some of this maybeoccurring(moreimportant near the strong network patches; see Lites et al. 1999, but note that we avoid these places anyway), the very different shapes between the granular and lane profiles indicate an intrinsically different origin. Higher spatial resolution would only increase the differences found here between the granules and the lanes. Another explanation for the granular signals could be that they are the result of observing the canopies of intergranular fields expanded on top of the granulation. However, we can also reject this explanation. For this, we return to the strong positive asymmetries observed in the granular fields. We have already mentioned that the canopy effect, when in conjunction with a downflow, produces positive asymmetries. Thus, when the canopy is observed on top of an upflow, it should produce very strong negative asymmetries, which is the opposite of what is observed. Instead, we propose that the granular signals are an intrinsic component of the upflowing pattern of granulation. Indeed, the presence of granular fields at very low signal levels is not a new result of this work. As shown by the IR work of Khomenko et al. (2003), a similar number of pixels with magnetic fields is found in granules and lanes, with a slight tendency for granular fields to be weaker (cf. their Fig. 8). Upflow regions and L1 signals in the downflow areas are at equipartition values or weaker (as indicated by the ratios between the amplitudes of the two Fe i lines analyzed in this paper). We suggest that these fields represent the reservoir of weak fields for the processes that eventually generate kg fields in the solar atmosphere, most likely by means of convective collapse (see, e.g., Solanki 1993) or some other form of magnetic concentration. 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