Topological changes of the photospheric magnetic field inside active regions: A prelude to flares?

Topological changes of the photospheric magnetic field inside active regions: a prelude to flares? Luca Sorriso-Valvo, Vincenzo Carbone, and Pierluigi Veltri Dipartimento di Fisica and Istituto Nazionale di Fisica per la Materia, Ponte P. Bucci, Cubo 31C, Universitá della Calabria, I-87036 Rende (CS), Italy∗ Valentina I. Abramenko Crimean Astrophysical Observatory, 98409 Nauchny, Crimea, Ukraine† Alain Noullez and Helene Politano Laboratoire Cassini, Observatoire de la Côte d’Azur, B. P. 4229, F-06304 Nice Cedex 04, France Annick Pouquet ASP/NCAR, P.O. Box 3000 Boulder, CO 80307-3000 Vasyl Yurchyshyn Big Bear Solar Observatory, 40386 North Shore Lane, Big Bear City, CA 92314-9672, U.S.A. 1 Abstract The detection of magnetic field variations as a signature of flaring activity is one of the main goal in solar physics. Past efforts gave apparently no unambiguous observations of systematic changes. In the present study, we discuss recent results on observations that scaling laws of turbulent current helicity inside a given flaring active region change in response to large flares in that active region. Such changes can be related to the evolution of current structures by a simple geometrical argument, which had been tested using high Reynolds number direct numerical simulations of the MHD equations. Interpretation of the observed data within this picture indicates that the change in scaling behavior of the current helicity seems to be associated with a topological reorganization of the footpoint of the magnetic field loops, namely with the dissipation of small scales structures in turbulent media. ∗ Electronic address: sorriso@fis.unical.it † Big Bear Solar Observatory, 40386 North Shore Lane, Big Bear City, CA 92314-9672, U.S.A. 2 I. INTRODUCTION Solar flares are sudden, transient energy release processes in active regions on the Sun (Priest, 1982). As a consequence of random motion of the footpoints of the magnetic field loops in the photospheric convective media (Parker, 1988), flares represent the dissipation of magnetic energy at numerous tangential discontinuities arising spontaneously in magnetic fields of active regions. The magnetic energy is released in various form as thermal, kinetic, soft and hard X-ray, accelerated particles etc. The observations of magnetic field variations, as a signature of flares in active regions, has been one of the main goals in solar physics, and some attempts for this have been made in the past (e.g. Hagyard et al., 1999 and references therein). All efforts give apparently no unambiguous observations of changes. Recently, observations of changes have been reported by Yurchyshyn et al., 2000. The authors observed systematic changes of the scaling behavior of the current helicity calculated inside an active region of the photosphere, connected to the eruption of large flares above that active region. In the present paper we conjecture that the changes in the scaling behavior of such observed quantity are related to the occurrence of changes in the topology of the magnetic field at the footpoint of the loop. II. CANCELLATION ANALYSIS AND TOPOLOGY OF TURBULENT STRUC- TURES The occurrence of scaling of signed measures, calculated from scalar fields f (x) which oscillate in sign, can be studied by using the cancellation analysis. For a given scalar field f (x), with x ∈ Q(L) where L is the size of the domain, we will introduce a coarse-graining of non overlapping boxes Qi (r) of size r, covering the whole domain Q(L). For each box Qi (r) the signed measure µi(r) can be defined as µi (r) = ! Qi (r) f (x)dx . (1) where i changes from 1 to the number of boxes Qi (r) needed to cover Q(L) at the scale r. The box size r is then a parameter representing a typical scale length. A partition function χ(r) can also be introduced by summing the signed measures (1) over all boxes covering the 3 region Q(L) at a given scale r χ(r) = " |µi (r)| (2) Qi (r) It had been observed that for fields presenting self-similarity, this quantity displays well defined scaling laws (Ott et al., 1992). That is, in a range of scales r, the partition function follows a power-law behavior χ(r) ∼ r −κ . (3) The scaling exponent κ is called a cancellation exponent (Ott et al., 1992) because it represents a quantitative measure of the scaling behavior of the imbalance between negative and positive contributions in the measure. For a positive definite measure or for a smooth field κ = 0, while κ = d/2 for a completely stochastic field in a d-dimensional space (for example a field of uncorrelated points with f = ±1, the sign being chosen randomly and independently for each point, with probability 1/2). As the cancellations between negative and positive parts of the measure decreases toward smaller scales, κ becomes positive. It is clear that the presence of various structures in a scalar field has an important effect on the cancellation exponent. For example, values of κ < d/2, (in the present paper d = 2), indicate the presence of sign-persistent (i. e. smooth) structures. In turbulent flows, the value of the cancellation exponent can be related to the characteristic fractal dimension D of turbulent structures on all scales using a simple geometrical argument (Sorriso-Valvo et al., 2002). Let λ be the typical correlation length of these structures, of the order of the Taylor micro-scale (see for example Frisch, 1995). In this case a scalar field is smooth (correlated) in D dimensions with a cutoff scale λ, and uncorrelated in the remaining d − D dimensions. In this sense, the dimension D represents the correlation dimension of the field. If the field is homogeneous, the partition function (2) can be computed as the number of boxes of size r, namely (L/r)d , times the integral over a single generic box Q(r) χ(r) = #$ # " ## Qi (r) dx f (x) ## #$ # # dr|f (x)| # Qi (r) Q(L) 1 ∼ d L frms % L r &d ! | Q(r) dx f (x)| . (4) The scaling of χ(r) can then be estimated by integrating the signed measure µ(r) over smaller regular sub-boxes of size λd recovering the box Q(r). The integration of the field over each sub-box of size λ in which the field is smooth returns the r.m.s. value of the field. The number of contributing sub-boxes can be estimated by considering separately 4 the correlated dimensions of the field and the uncorrelated ones. Integration over correlated dimensions will bring a contribution proportional to their area (r/λ)D , while the uncorrelated dimensions will contribute as the integral of an uncorrelated field, that is proportional to the square root of their area (r/λ)(d−D)/2 . Thus, when homogeneity is assumed, collecting all the contributions in (2) leads for the partition function to λd frms χ(r) ∼ d L frms % L r &d % &D % & d−D r λ r λ 2 ∼ % &− d−D r λ 2 ∼ % &−κ r λ (5) so that one can recover the simple relation κ = (d − D)/2 . III. (6) CANCELLATION ANALYSIS OF ACTIVE REGION MAGNETOGRAMS To get a quantitative measure of a change in the scaling of current helicity inside active regions, we used observations of the vector magnetic field obtained with the Solar Magnetic Field Telescope of the Beijing Astronomical Observatory (China). Measurements were recorded in the FeI 5324.19 Å spectral line. The field of view is about 218” × 314”, corresponding to 512 × 512 pixels on a CCD. The magnetic field vector at the photosphere has been obtained by measuring four Stokes parameters. The current density Jz (x, y) was calculated as a contour integral of the transverse field over a closed contour of size 1.72” × 1.86” (cf. Yurchyshyn et al., 2000, for details). The current helicity, Hc = B·J, where B represents the magnetic field and J = ∇ × B is the current density, is a measure of small-scale activity in magnetic turbulence. It indicates the degree of clockwise or counter-clockwise twisting of current structures. Let us consider a photospheric magnetogram of size L taken in an active region and let B⊥ (x, y) be the observed transverse magnetic field perpendicular to the line of sight, where (x, y) are coordinates on the surface of the sun. By using this transverse field, we can calculate the restriction of the current helicity to the z-components of the fields, or in short the z-related part of current helicity, namely hc (x, y) = Bz (x, y)Jz (x, y), where Jz (x, y) = [∇ × B⊥ ] · êz . For simplicity, in this paper we will in fact refer to the z-related part of current helicity as current helicity. Figure 1 shows the current helicity calculated for an active region NOAA 7590 for a given time, which is close to the flaring time. The presence of signed structures is clearly visible. 5 A signed measure can be defined from the current helicity as follows: µi (r) = ! Qi (r) hc (x, y)dxdy . (7) In Figure 2 we show, as an example, the scaling behavior of χ(r) vs. r for a flaring active region NOAA 7315, which started to flare on October 22, 1992. At the large scales we find χ(r) ∼ const, which is due to the complete balance between positive and negative contributions. If the resolution of the magnetograms is high enough to resolve small structures of current helicity, the partition function shown in Figure 2 would saturate at the small scales. This is not seen in the data, indicating the presence of structures smaller than the resolution scale of the magnetograms. In the intermediate range of scales, the cancellation exponent is found to be 0.53 ± 0.09 (Yurchyshyn et al., 2000). Let us now consider how the fractal dimension of current structures D changes as a function of time. To this aim, we take a set of consecutive magnetograms of the same active region and for each magnetogram we compute the values of κ and D by using relation (6). Note that, since cancellations in the vertical photospheric magnetic field Bz (x, y) had been found to be very small (Lawrence et al., 1993; Abramenko et al., 1998a), with a cancellation exponent of the order of 10−2 , cancellations in the current helicity are entirely due to current structures. As an example, in Figure 3, we show the time evolution of D compared to the X-ray flux measured in an active region NOAAs 7590. We observe that the fractal dimension D, becomes abruptly large in correspondence with a sequence of large (C and M class) flares, with flux above 10−6 W/m2 , occurring in the corona. The same behavior has been found for all calculations in all active regions with major flares we examined (more examples are reported in Abramenko et al., 1998a; Yurchyshyn et al., 2000; Abramenko, 2002). The detailed analysis of the cancellation exponent variations is reported in Yurchyshyn et al. (2000). The values of the fractal dimension D for five active regions are given in Table 1, together with the occurrence of C-class or larger flares. ¿From this analysis, it seems that the fractal dimension D presents jumps only if major flares are observed. In fact, in a very quiet active region (see AR7216 values in Table 1), variations of D are of the order of 10%, against the 30-40% observed in all other active regions with large flares. Apart from this, no correlation has been found between the value of the jump of D and the intensity of the associated sequence of large flares. 6 As we have already mentioned, as the density of the measure becomes smooth (no changes in sign are present) we may see a saturation of χ(r). As shown in Figure 2, saturation of χ(r) is observed at large scales. The fact that we do not find this saturation at the small scales is indirect evidence that elementary flux tubes are smaller than the spatial resolution of the data. The change toward large D, indicating smoothing of small-scale current structures, is then, probably, due to small-scale dissipation of magnetic energy. That is, magnetic energy is suddenly transferred toward small scales, as a manifestation of a turbulent energy cascade. IV. CANCELLATION ANALYSIS OF NUMERICAL DATA The model linking the fractal dimension of structures with the cancellation exponent can be tested using numerical data. We point out that the numerical data used in this Section is not supposed to simulate the complexity of actual photospheric plasma. However, we use such data in order to check the validity of the model (6) in a simpler context. Using high resolution turbulent fields, obtained from two-dimensional (d = 2), periodic, incompressible, forced magnetohydrodynamic simulations (Politano et al., 1998; Sorriso-Valvo et al., 2000), we can build-up different signed measures. For example, since the geometry of the magnetic field B(x, y) = (Bx , By , 0) is two-dimensional, the current J (x, y) = ∇ × J = (0, 0, Jz ) has only a z component. In Figure 4 we display the electric current field J(x, y), obtained from our numerical data. The gray-scale map shows one snapshot of the field in a statistically steady state, during a time interval starting at t = 168 until t = 336. Note that time is measured in non-linear time units, τN L , based on the rms velocity and the integral scale. As in the case of the solar data, the presence of positive and negative current structures is evident in the simulated image (Figure 4). The signed measure of electric current can be then computed as: µi (r) = ! Qi (r) Jz (x, y) dx dy , and the scaling properties of the partition function are reported in Figure 5. Note that the partition function presented in Figure 5 is computed as the time average in the statistically steady time interval mentioned above. The power-law scaling (3) is clearly visible in a spatial range extending from the large scales (near the integral scale of the flow %0 ∼ 0.2L, L = 2π being the size of the simulation box) down to a correlation lenght r ⋆ of the order of 7 the Taylor micro-scale of the flow λ ∼ 0.02L (see for example Frisch, 1995). In this region, we fit the partition function and obtained the cancellation exponent κ = 0.43 ± 0.06. A saturation of the partition function is observed at a scale rS which is found to be of the order of the dissipative scale of the flow. In fact, for scales smaller than rS the dissipation stops the cascade of formation of small scale structures, therefore cancellations end too. Fractal dimension of current structures has been computed using equation (6), which gives D ≃ 1, indicating that current structures are similar to filaments. The presence of current filaments can be clearly observed by a direct inspection of the current field contour plot, confirming the reliability of the model (see Sorriso-Valvo et al., 2002). In order to directly compare the numerical results with the solar data cancellation analysis, we calculate the current helicity for the modeled field. As mentioned before, in twodimensional geometry, the electric current is perpendicular to the magnetic field, hence the current helicity is zero; we thus simply consider Hc(2d) (x, y) = J(x, y)|B(x, y)|, which is shown in Figure 6 for the same time as in Figure 4. The modeled current helicity, as in the case of the solar data, appears smoother than the electric current map, and has the same topology of structures. The signed measure of such a field is then computed as in previous cases: µi (r) = ! Qi (r) Hc(2d) (x, y) dx dy . The scaling properties of the partition function χ(r) can now be represented by a cancellation exponent, obtained by the usual fitting procedure after time averaging. In Figure 7 we present the scaling of χ(r) along with the power-law fitting with the power index κ = 0.46 ± 0.03. This index is very close to that obtained for the electric current; this shows that in the case of the two-dimensional numerical simulations presented here, the current field is the one responsible for sign singularities, and the current structures control the cancellations. However, it remains to be confirmed, e. g. with the help of three-dimensional numerical simulations, whether indeed all such correlation functions built on the magentic field and its derivatives have or not identical scaling laws. We want now to consider in more detail the time evolution of cancellation effects in the two-dimensional numerical simulations. To do this, we plot in Figure 9 the time evolution of the (kinetic) Reynolds number, together with the two cancellation exponents κj and κH (2d) , c for the snapshots already presented in Figures 4 and 6. Moreover, in Figure 8 we plot ten snapshots of the current helicity field Hc(2d) , taken within the time interval from t = 168 until 8 t = 336. The first three snapshots look smoother than the following ones, and this fact can be interpreted as a stronger presence of dissipative effects at such times; correspondingly, the cancellation exponents are smaller in the first three snapshots than in the following ones. This would mean, following our model, that the fractal dimension of the structures D is larger for these snapshots, as qualitatively confirmed by the smoother aspect of the field at such time (Figure 8). This observation confirms that the cancellation exponent is linked to the topology of the field, as already observed by Sorriso-Valvo et al (2002). The time evolution of the Reynolds number is shifted (time-lag) with respect to the evolution of the cancellation exponents. Unfortunately, due to the limited time interval of our simulations, it is impossible to say whether that shift is backward or forward. Since the Reynolds number is related to the importance of dissipative effects against non-linear effects in the turbulent cascade, it would be interesting to clarify this question as a further confirmation of our interpretation. This point is left for future work. V. CONCLUSIONS In this paper we build a model which allows us to recognize changes in the behavior of the photospheric magnetic field of active regions. These changes, detected by variations of a scaling index for turbulent cancellations of the current helicity, are mainly observed before the start of a sequence of large (C-class or larger) flares occurring in the corona. The variations of the scaling index are due to the topology changes of the structures present in the magnetic field, and, thus, are related to the non-linear, intermittent turbulent cascade, underlying the formation of such structures. This suggestion has been tested using high resolution numerical data. The topological changes of the magnetic structures are linked to small-scale dissipation. Thus, our picture supports the idea of E. Parker concerning an avalanche of small reconnection events as the main cause of solar flares (see Parker, 1987; see also Abramenko et al., 1998a). Recently, an analysis of the intermittency variations during flares by Abramenko et al., 1998a) also supported this picture. However, we have to point out that the temporal resolution of the solar data is poor. Unfortunately, this kind of observations needs several restrictive conditions, namely a strong flare, an active region near the solar disk center, telescope day-time at an observatory, good seeing, several magnetograms before the flare and, at least, one at the flare maximum, 9 conditions which are seldom met. Nevertheless, the behavior observed for the five cases reported in Table 1, as well as in the other cases found in the literature (Abramenko et al., 1998b) suggests that the preliminary result presented here is significant enough to encourage a search for similar correlations, and to explore more thoroughly the fine structure of flares, in order to be able to obtain all the necessary information leading to the computation of the fractal dimension D, and, thus, to improve the temporal resolution of the observations. To conclude, the analysis presented here indicates what should be a candidate signature of the occurrence of large flares, the cancellation analysis giving informations about the physical statistical processes underlying flares. It could then be considered as a useful tool in the attempt to forecast the occurrence of strong flaring activity in active regions. 10 [1] Abramenko, V.I., Yurchyshyn V.B., and Carbone, V., Does the photospheric current take part in the flaring process?, Astron. Astrophys., 334, L57, 1998. [2] Abramenko, V.I., Yurchyshyn V.B., and Carbone, V., Sign-singularity of the current helicity in solar active regions, Solar Phys., 178, 35, 1998b. [3] Abramenko, V.I., Solar MHD Turbulence in Regions with Various Levels of Flare Activity, Astron. rep. 46, 161, 2002, [4] Frisch, U., Turbulence, Cambridge University Press, 1995. [5] Hagyard, M.J., Stark, B.A., and Venkatakrishnan, P., A search for vector magnetic field variations associated with the M-class flares of 10 June 1991 in AR 6659, Sol. Phys., 184, 133, 1999. [6] Lawrence, J.K., Ruzmaikin, A.A., and Cadavid, A.C., Astrophys. J., 417, 805, 1993 [7] Ott, E., Du, Y., Sreenivasan, K.R., Juneja, A., and Suri, A.K., Phys. Rev. Lett., 69, 2654, 1992. [8] Parker, E.N., Stimulated dissipation of magnetic discontinuities and the origin of solar flares, Solar Phys.., 111, 297, 1987. [9] Politano, H., Pouquet, A., and Carbone, V., Determination of anomalous exponents of structure functions in two-dimensional magnetohydrodynamic turbulence, Europhys. Lett., 43, 516, 1998. [10] Priest, E., Solar Magnetohydrodynamic, D. Reidel Publishing Company, Dordrecht, 1982. [11] Sorriso-Valvo, L., Carbone, V., Noullez, A., Politano, H., Pouquet, A., and Veltri, P., Analysis of cancellation in two-dimensional MHD turbulence, Phys. Plasmas, 9, 89, 2002. [12] Sorriso-Valvo, L., Carbone, V., Veltri, P., Politano, H., and Pouquet, A., Non-gaussian probability distribution functions in two-dimensional magnetohydrodynamic turbulence, Europhys. Lett., 51, 520, 2000. [13] Yurchyshyn, V.B., Abramenko, V.I., and Carbone, V., Flare-related changes of an active region magnetic field, Astrophys. J., 538, 968, 2000. 11 FIG. 1: The current helicity Hc measured for the active region NOAA 7590 on October 3, 1992. The presence of positive and negative structures on all scales is clearly visible. The flat portion of field near the corners are due to removing in the projection effects. 12 3.0 2.5 =0.53 (r) 2.0 1.5 NOAA 7315 0.2 0.3 0.4 0.5 0.6 0.7 r/L FIG. 2: An example of the scaling of the partition function for a flaring active region (NOAA 7315), which started to flare on October 22, 1992. The power-law fit is indicated as a dotted line. 13 10-3 0.8 NOAA 7590 10-4 0.7 M -5 10 D C 0.6 10-6 Flux (Watt/m2) X B 0.5 0 -7 20 40 10 60 Time (hours) FIG. 3: Flare intensity in Watt/m2 recorded in the active region NOAA7590, including one large M-class flaring event, observed October 1-3, 1993 (line bars, right vertical axis). The corresponding time variation of the fractal dimension D is reported with symbols linked by a dash line (left vertical axis). 14 FIG. 4: Current field J obtained from a high resolution two-dimensional numerical simulation of the MHD equations. The plot refers to one snapshot in the statistically steady state, at t = 260 in non-linear times units, τN L . As in the case of the solar data, the presence of positive and negative structures on all scales is clear. 15 1 10 =0.43 100 -1 10 (r) J 10-2 -3 10 -3 10 10-2 10-1 100 r/L FIG. 5: Scaling of the partition function for the current stemming from numerical data. This result is obtained by averaging the time evolution, in order to increase the quality of the statistics. A power-law fit is indicated as a straight line. The scales are normalized to the simulation box size L = 2π. 16 (2d) FIG. 6: Current helicity Hc obtained from a high resolution two-dimensional numerical simula- tion of the MHD equations (same snapshot as that presented for the current in Figure 4). Signed structures are present and reproduce the current structures, but the current helicity field looks smoother than the current field itself. 17 100 Hc (r/L) 10 = 0.46 -1 10-2 -3 10 10-2 10-1 100 r/L FIG. 7: Scaling of the partition function for the current helicity stemming from numerical data. As for the current, this result is obtained by averaging the time evolution, in order to increase the statistics. The power-law fit is indicated as a straight line. The scales are normalized to the simulation box size L = 2π. 18 (2d) FIG. 8: Time evolution of the current helicity field Hc , obtained from a high resolution two- dimensional numerical simulation of the MHD equations with ten snapshots in the statistically steady state. For each snapshot, the values of the cancellation exponent κ and the fractal dimension D are reported. The frames are dropped for clarity. 19 0.7 1800 0.6 1600 1400 Re (t) 0.5 1200 0.4 J Hc (v) e R 0.3 168 (v) 196 224 252 t/ 280 308 1000 336 NL FIG. 9: Cancellation exponents κ for both the current (black circles) and the current helicity (stars) for different times (left vertical axis). The plot with open circles represents the kinetic Reynolds number of the flow (right vertical axis). 20 TABLE I: For five active regions, we report the values of the fractal dimension D of the photospheric magnetic structures, as computed from the cancellation exponents, following our model (see text). The start times and the classes of large flaring events occurring in the corresponding active regions, are also indicated. All times are measured in hours, starting from the first magnetogram for each AR (taken at t = 0.00). Note that the first active region does not present large flares. AR time (h) flare D 7216 0.00 1.10 2.55 3.68 4.85 — — — — — 0.894 ± 0.020 0.986 ± 0.006 0.902 ± 0.020 0.952 ± 0.010 1.000 ± 0.016 7315 0.00 — — 22.78 — 23.02 25.05 47.45 48.67 51.02 — 20.75 (C) 21.62 (C) — 22.82 (M) — — — — — 0.952 ± 0.008 — — 0.920 ± 0.012 — 1.170 ± 0.014 1.300 ± 0.008 1.286 ± 0.008 1.256 ± 0.012 1.090 ± 0.010 7590 0.00 26.50 — 27.35 — 28.92 51.98 — — 27.18 (M) — 28.77 (M) — — 0.476 ± 0.014 0.418 ± 0.020 — 0.954 ± 0.004 — 0.892 ± 0.008 0.900 ± 0.006 7585 0.00 — 3.37 3.52 — 4.37 5.75 23.53 — 2.53 (C) — — 4.10 (B) — — — 0.776 ± 0.012 — 1.178 ± 0.010 1.218 ± 0.008 — 1.108 ± 0.012 0.978 ± 0.012 0.892 ± 0.012 6891 2.38 0.47 1.28 2.63 — — 3.47 4.45 — — — — 2.95 (C) 3.25 (X) — — 1.008 ± 0.006 0.932 ± 0.006 0.672 ± 0.010 0.886 ± 0.006 — — 0.930 ± 0.008 0.746 ± 0.008 21