The definition of the fractal dimension D _A for 2-D areas A is also called the Hausdorff dimension D _A0 (Mandelbrot 1977), A 0 = L 0 D A 0 , (1) or explicitly (normalized at i = 0), D A 0 = log A 0 log L 0 , (2) where the area A ₀ is the sum of all image pixels I (x, y) ≥ I ₀ above a background threshold I ₀, and L ₀ is the length scale of a fractal area. A structure is fractal, when the ratio D _Ai is approximately constant versus different length scales L _i and converges to a constant for the smallest length scales L↦0. The method described here is also called the box-counting method, because the number of pixels are counted over an area A ₀ and length scale L ₀.

In analogy, a fractal dimension can also be defined for the 3-D volume V, V 0 = L 0 D V 0 , (3) or explicitly D V 0 = log V 0 log L 0 , (4) The valid range for these two area fractal dimensions is 1 ≤ D _A ≤ 2 and 2 ≤ D _V ≤ 3, where D = 0, 1, 2, 3 are all possible Euclidean dimensions.

We can estimate the numerical values of the fractal dimensions D _A and D _V from the means of the minimum and maximum values in each Euclidean domain, D A = D A , min + D A , max 2 = 1 + 2 2 = 3 2 = 1.50 , (5) and correspondingly, D V = D V , min + D V , max 2 = 2 + 3 2 = 5 2 = 2.50 . (6) The 2-D fractal dimension D _A is the most accessible SOC parameter, while the 3-D fractal dimension D _V requires information of fractal structures along the line-of-sight, either using a geometric or tomographic model, or modeling of optically-thin plasma (in the case of an astrophysical object observed in soft X-ray or EUV wavelengths).

We find that the theoretical prediction of D _A = (3/2) = 1.50 (Eq. (5)) for the fractal area parameter A is approximately consistent with the observed values obtained with the box-counting method, D A obs = 1.54 ± 0.04 (Table 1).

The size distribution N(L) of length scales L, also called the scale-free probability conjecture (Aschwanden 2012; Aschwanden 2014), which essentially is the standard expression for the probability conservation in a power law distribution, N L d L ∝ L − d d L , (7) where d is the Euclidean space dimension, generally set to d = 3 for most real-world data. Note, that this occurrence frequency distribution function is simply a power law, which results from the reciprocal relationship of the number of events N(L) and the length scale L. Since the fractal dimension D _A for event areas A is defined as (Eq. 1), A = L D A , (8) we obtain the inverse function L(A), L = A 1 / D A , (9) and the derivative, d L d A = A 1 / D A − 1 , (10) so that we obtain the area distribution N(A) by substitution of L (Eq. 9) and the derivative dL/dA (Eq. 10) into N(L) (Eq. 7), N A d A = N L A d L d A d A = L A − d A 1 / D A − 1 d A = A − α A d A , (11) which yields the power law index α _A , for d = 3, α A = 1 + d − 1 D A = 1 + 2 D A . (12) Vice versa we can then obtain the SOC fractal dimension D _A from an observed power law slope α _A (Table 1), by inverting α _A  (D _A ) in Eq. 12, D A SOC = 2 α A − 1 = 3 2 . (13) Using the theoretical value α _A = 7/3 ≈ 2.33 (Table 2), we expect a value of D A SOC = 1.5 (Eq. 13), which is identical with the prediction of the mean Euclidean dimension D A M E = 1.5 (Eq. 5) based on the mean of the extremal maximum and minimum values. This is an alternative method (Eq. 13) to calculate the fractal area dimension, in contrast to the mean Euclidean method (Eq. 5), which we call the SOC-inferred fractal dimension, because it uses the (power law) size distribution of areas that are defined in SOC models.

Our first measurement is the fitting of a power law distribution function N ( A ) ∝ A − α A of the event (or nanoflare) areas A, separately for each of the 12 IRIS datasets, as shown in Figure 1. The area of the event is a combination of all the spatially connected 3-σ pixels throughout their lifetime. The lowest bin was discarded in the histogram when a visible deviation from a power law was apparent. The number of events amounts to 23,633 for all 12 datasets together, varying from 65 to 4,725 events per IRIS dataset (Table 1). The power law slope fits vary from the lowest value α _A = 2.14 (dataset #1) to the highest value α _A = 2.83 (dataset #4), having a mean and standard deviation of (Figure 2, top panel). a A obs = 2.51 ± 0.21 . (14) The area size distributions are shown superimposed for the 12 IRIS datasets (Figure 2, top panel), which illustrates almost identical power law slopes in different IRIS datasets.

Fitting the energy size distributions, N ( E ) ∝ E − α E , yields the following mean for all 12 IRIS datasets (Figure 2, middle panel), a E obs = 2.03 ± 0.18 . (15)

Fitting the duration size distributions, N ( T ) ∝ T − α T , yields the following mean for all 12 IRIS datasets (Figure 2, bottom panel), a T obs = 2.65 ± 0.39 . (16) We will interpret these power law slopes in terms of SOC models in Size distributions.

The next parameter that we are interested in is the fractal dimension. A standard method to determine the fractal dimension D _A of an image is the box-counting method, which is defined by the asymptotic (L↦0) ratio of the fractal area A to the length scale L, i.e., D _A = log(A)/log(L), also called Hausdorff (fractal) dimension. We test the fractality by varying the pixel sizes (or spatial resolution) by powers of two, i.e., L _i = 2 ⁱ = [1, 2, 4, 8] for i = [0, 1, 2, 3]. In order to normalize to the same number of events for each spatial resolution, the fractal (Hausdorff) dimension is defined by (e.g., Hirzberger et al., 1997), D A , i = log A i 2 2 i log L i 2 i , (17) where L _i is the observed length scale, and A _i is the observed fractal area, measured at different spatial resolutions. If the fractal dimension D _A,i stays more or less constant for different spatial resolutions L _i = [1, 2, 4, 8], then the dimension D _A,i is said to be “fractal”.

It has been pointed out that the detection of small-scale impulsive events requires a careful subtraction of event-unrelated background noise in the IRIS 1400 Å data (Vilangot Nhalil et al., 2020). The main effect of background subtraction is the related change in the fractal area, which causes a sensitive bias: If too much background is subtracted, the fractal area is smaller and the resulting fractal dimension is too small, and vice versa when the estimated background is under-estimated. At times and locations where no impulsive events occur, the flux distribution of an image shows a Gaussian distribution function (due to the random noise), while a heavy-tail occurs during active times (due to SOC-generated avalanches). In the case of a dominant noise component, a Gaussian can be fitted to the size distribution function, which yields a mean I _avg and a standard deviation I _sig. A 1-σ threshold can then be defined by, I thr = I avg + I sig . (18) which separates the linear noise fluctuations (at I (x, y) ≤ I _thr ) from the nonlinear avalanches (at I (x, y) ≥ I _thr). The calculation of a fractal dimension is then obtained from the ratio log (A _i 2²ⁱ )/log (L _i 2 ⁱ ) (Eq. 17), where the area A _i includes a count of all pixels with a flux value above the threshold, i.e., I (x, y) > I _thr, and the length scale L _i is the number of pixels that measure the length scale of a SOC avalanche.

We show the fractal dimensions measured with Eq. 17, for each of the 12 IRIS datasets and the 4 spatial resolutions D _A0, D _A1, D _A2, and D _A3 in Table 1, which reveal a very narrow spread of values D _A for the fractal dimension, with a mean and standard deviation of a few percents (Table 1), D A obs = 1.55 ± 0.07 . (19) Note, that the values obtained from different IRIS datasets and with different spatial resolutions are all consistent among each other and do not show any systematic dependency on the spatial resolution. Moreover, they are consistent with the theoretical expectation of the mean Euclidean dimension D A M E (Eq. 5) and the SOC-Inferred value D A SOC (Eq. 13), D A M E = D A SOC = 3 2 = 1.5 . (20) The fractal nature of the 12 IRIS datasets is rendered in Figure 3 and 4, where the black areas correspond to zones with enhanced emission, and the white areas correspond to the background with weak emission. The successive reduction of spatial resolution is shown in Figure 4.

An example of a theoretical fractal pattern with a close ressemblance to the observed transition region patterns of dataset #8 is shown in Figure 5, which is called the “golden dragon fractal” and has a Hausdorff dimension of D _A = 1.61803.

In Table 3 we compile fractal dimensions obtained from photospheric, chromospheric, and transition region fractal features, which may be different from coronal and flare-like size distributions. The fractal dimension has been measured in photospheric wavelengths with the perimeter-area method, containing dominantly granules and super-granulation features (Roudier and Muller 1986; Hirzberger et al., 1997; Berrilli et al., 1998; Bovelet and Wiehr 2001; Paniveni et al., 2005, 2010), which exhibit a mean value of (Table 3), D A gran = 1.23 ± 0.09 . (21) Although this mean value is averaged from different solar features (granular cells and supergranular cells), as well as from different atmospheric heights (photospheric and chromospheric Ca II K data), the fractal dimension varies only by a small factor of ±7%. We have to be aware that photospheric emission originates from a lower altitude than any transition region or coronal feature. The relatively low value obtained for granulation features thus indicates that the granulation features seen in optical wavelengths are almost curvi-linear (with little area-filling topologies), which is expected for sparse photospheric mass flows along curvi-linear flow lines.

A second feature we consider are plages in the transition region, measured with IRIS 1400 Å (Vilangot Nhalil et al., 2020), which have formation temperatures of ≈ 1 0 3.7 − 1 0 5.2 K in the lower transition region, exhibiting a mean value of (Table 3), D A plage = 1.54 ± 0.04 . (22)

A third feature is an active region, observed in photospheric magnetograms and analyzed with the linear-area method (Lawrence 1991; Balke et al., 1993; Lawrence and Schrijver 1993; Meunier 1999; Janssen et al., 2003; Meunier 2004; Ioshpa et al., 2008), or with the box-counting method (McAteer et al., 2005). The mean value of fractal dimensions measured in active regions is found to be (Table 3), D A A R = 1.59 ± 0.20 . (23) Apparently, active regions organize magnetic features into quasi-space-filling, area-like geometries.

Nanoflare events constitute a fourth phenomenon, which has been related to the SOC interpretation since Lu and Hamilton (1991). Nanoflares have been observed in EUV 171 Å and 195 Å with the TRACE instrument, as well as in soft X-rays using the Yohkoh/SXT (Solar X-Ray Telescope) (Aschwanden and Parnell 2002), which show a mean value of (see Table 3), D A nano = 1.56 ± 0.08 . (24) Nanoflares have been observed in the Quiet Sun and appear to have a similar fractal dimension as impulsive brightenings in active regions measured in magnetograms.

For completeness we list also the fractal dimension measured in large solar flares, for M-class flares, X-class flares, and the Bastille Day flare (Aschwanden and Aschwanden 2008a), as observed in the EUV, which all together exhibit a mean value of (Table 3), D A flare = 1.76 ± 0.14 . (25) This is the largest mean value of any measured fractal dimension, which indicates that the flare process fills the flare area almost completely, due to the superposition of many coronal postflare loops that become filled as a consequence of the chromospheric evaporation process.

Thus, we can distinguish four groups with significantly different fractal properties in photospheric, chromospheric, transition region, and coronal data (Table 4). A first group has a very low fractal dimension (D _A ≈ 1.2) that indicates curvi-linear features produced by super-granulation flows, a second group with chromospheric (network) features has a mean of (D _A ≈ 1.4), a third group with intermediate fractal dimensions (D _A ≈ 1.54) includes active region features in the photosphere, plages in the transition region, and EUV nanoflare events, and a fourth group with high values of fractal dimenions (D _A ≈ 1.8) that includes large (M- and X-class) flares, likely to be caused by area-like topologies of magnetic reconnection and chromospheric evaporation processes.

A fractal geometry is a ratio that provides a statistical index of complexity, and changes as a function of a length scale that is used as a yardstick to measure it (Mandelbrot 1977). There are four integer values of Euclidean dimensions d = [0, 1, 2, 3]: zero-dimensional point-like structures (d = 0), one-dimensional linear or curvi-linear structures (d = 1), two-dimensional area-like structures (d = 2), and three-dimensional voluminous structures (d = 3). All other values between 0 and 3 are non-integer Euclidean dimensions and are called fractal dimensions.

Basic methods to measure fractal dimensions include the linear-area (LA) method, the perimeter-area (PA) method, and the box-counting (BC) method. The LA method calculates the ratio of a fractal area A to a quasi-space-filled (encompassing) quadratic area with size L ². Similarly, the PA method yields a ratio of the encompassing curve length or perimeter length (P = πr in the case of a circular boundary). The box-counting method uses a cartesian (2-D or 3-D) lattice grid [x, y] and counts all pixels above some threshold or background, and takes the ratio to the total counts of all pixels inside the encompassing coordinate grid. These three methods appear to be very simple, but are not unique. The resulting fractal dimensions may depend on the assumed level of background subtraction, or on the spatial resolution, if not properly normalized. The encompassing perimeter depends on the definition of the perimeter (square, circle, polygon, etc.). Multiple different geometric patterns may cause a variation of the fractal dimension across an image or data cube. Temporal variability can modulate the fractal dimension as a function of time. Detailed discussions and examples of the topics of the background subtraction, the spatial resolution, the selection of the field-of-view, and the temporal stability are discussed in almost all references that are listed in Table 3. The detailed incorporation of a fractal measurement method differs in each study.

Theoretical values of fractal dimensions converge by definition to a unique value (e.g., D _A = 1.61803 for the golden dragon fractal, Figure 5), while observed data almost always exhibit some spatial inhomogeneity that gives rise to a spread of fractal dimension values across an image.

A compilation of fractal dimensions measured in photospheric, chromospheric, and coronal wavelengths is given in Table 3. The solar granulation has a typical spatial scale of L = 1,000 km, or a perimeter of P = πL ≈ 3000 km. Roudier and Muller (1986) measured the areas A and perimeters P of 315 granules and found a power law relation P ∝ A ^D/2, with D = 1.25 for small granules (with perimeters of p ≈ 500–4,500 km) and D = 2.15 for large granules (with p = 4,500–15, 000 km). The smaller granules were interpreted in terms of turbulent origin, because the predicted fractal dimension of an isobaric atmosphere with isotropic and homogeneous turbulence is D = 4/3 ≈ 1.33 (Mandelbrot 1977). Similar values (D _A = 1.30 and D _A = 1.16) were found by Hirzberger et al. (1997), Ermolli et al. (1998), and Berrilli et al. (1998). Bovelet and Wiehr (2001) tested different pattern recognition algorithms (Fourier-based recognition technique FBR and multiple-level tracking MLT) and found that the value of the fractal dimension strongly depends on the measurement method. The MLT method yielded a fractal dimension of D _A = 1.09, independent of the spatial resolution, the heliocentric angle, and the definition in terms of temperature or velocity. Paniveni et al. (2005) found a fractal dimension of D _A ≈ 1.25 and concluded, by relating it to the variations of kinetic energy, temperature, and pressure, that the super-granular network is close to being isobaric and possibly of turbulent origin. Paniveni et al. (2010) investigated super-granular cells and found a fractal dimension of D _A = 1.12 for active region cells, and D _A = 1.25 for quiet region cells, a difference that they attributed to the inhibiting effect of the stronger magnetic field in active regions. Averaging all fractal dimensions related to granular datasets we obtain a mean value of D _A = 1.23 ± 0.09, which is closer to a curvi-linear topology (D _A ≳ 1.0) than to an area-filled geometry (D _A ≲ 2.0).

The physical understanding of solar (or stellar) granulation has been advanced by numerical magneto-convection models and N-body dynamic simulations, which predict the evolution of small-scale (granules) into large-scale features (meso- or super-granulation), organized by surface flows that sweep up small-scale structures and form clusters of recurrent and stable granular features (Hathaway et al., 2000; Berrilli et al., 2005; Rieutord et al., 2008; Rieutord et al., 2010).

The fractal multi-scale dynamics has been found to be operational in the quiet photosphere, in a quiescent non-flaring state, as well as during flares (Uritsky and Davila 2012).

The fractal structure of the solar granulation is obviously a self-organizing pattern that is created by a combination of subphotospheric magneto-convection and surface flows, which are turbulence-type phenomena.

Measurements of the fractal dimension and power law slope of the size distribution in the transition region have been accomplished with IRIS 1400 Å observations of plages and sunspot regions (Vilangot Nhalil et al., 2020; and this work, see Table 3). Fractal dimensions of transition region features were evaluated with a box-counting method here, yielding a range of D _A ≈ 1.54 ± 0.04 for the 12 datasets of plages in the transition region listed in Tables 1 and 3. The structures observed in the 1,400 Å channel of IRIS are dominated by the Si IV 1394 Å and 1,403 Å resonance lines, which are formed in the transition region temperature range of T = 10^4.5–10⁶ K, sandwiched between the cooler chromosphere and the hotter corona. Apparently, the fractal dimension is not much different in plages with sunspots (D _A = 1.58 ± 0.05), or in field-of-views without sunspots (D _A = 1.52 ± 0.09), (Table 3).

One prominent feature in the transition region is the phenomenon of “moss”, which appears as a bright, dynamic pattern with dark inclusions, on spatial scales of L ≈ 1–3 Mm, which has been interpreted as the upper transition region above active region plages and below relatively hot loops (De Pontieu et al., 1999). Besides transition region features, measurements in chromospheric (Quiet-Sun) network structures in the temperature range of T = 10^4.5–10⁶ K yield fractal dimensions of D _A = 1.30–1.70 (Gallagher et al., 1998). Furthermore, a value of D _A ≈ 1.4 was found for so-called Ellerman bombs (Georgoulis et al., 2002), which are short-lived brightenings seen in the wings of the Hα line from the low chromosphere. In addition, a range of D _A ≈ 1.25–1.45 was measured from a large survey of 9342 active region magnetograms (McAteer et al., 2005). Measurements of SOHO/CDS in EUV lines in the temperature range of T _e ≈ 10^4.5–10⁶ revealed a distinct temperature dependence: fractal dimensions of D _A ≈ 1.5–1.6 were identified in He I, He II, OIII, OIV, OV, Ne VI lines at log (T _e ) ≈ 5.8, then a peak with D _A ≈ 1.6–1.7 at log (T _e ) ≈ 5.9, and a drop of D _A ≈ 1.3–1.35 at log (T _e ) ≈ 6.0 (see Figure 11 in Gallagher et al., 1998). The temperature dependence of the fractal dimension can be interpreted in terms of sparse heating that produces curvi-linear flow patterns with low fractal dimensions of D _A ≲ 1.5, while strong heating produces volume-filling by chromospheric evaporation with high fractal dimensions D _A ≳ 1.5.

In recent work it was found that the concept of mono-fractals has to be generalized to multi-fractals to quantify the spatial structure of solar magnetograms more accurately (Lawrence and Schrijver 1993; Cadavid et al., 1994; Lawrence et al., 1996; McAteer et al., 2005; Conlon et al., 2008; Giorgi et al., 2015).

A number of studies investigated the fractal dimension of the photospheric magnetic field, as observed in magnetograms in the Fe I (6,302 Å, 5,250 Å) or Ni I (6,768 Å) lines. Meunier (1999) evaluated the fractal dimension with the perimeter-area method and found D _A = 1.58 for super-granular structures to D _A = 1.58 for the largest structures, while the linear size-area method yielded D _A = 1.78 and D _A = 1.94, respectively. In addition, a solar cycle dependence was found by Meunier (2004), with the fractal dimension varying from D _A = 1.09 ± 0.11 (minimum) to D _A = 1.73 ± 0.01 for weak-field regions (B _m < 900 G), and D _A = 1.53 ± 0.06 (minimum) to D _A = 1.80 ± 0.01 for strong-field regions (B _m > 900 G), respectively. A fractal dimension of D _A = 1.41 ± 0.05 was found by Janssen et al. (2003), but the value varies as a function of the center-to-limb angle and is different for a speckle-reconstructed image that eliminates seeing and noise.

A completely different approach to measure the fractal dimension D was pursued in terms of a 2-D diffusion process, finding fractal diffusion with dimensions in the range of D ≈ 1.3–1.8 (Lawrence 1991) or D = 1.56 ± 0.08 (Lawrence and Schrijver 1993) by measuring the dependence of the mean square displacement of magnetic elements as a function of time. Similar results were found by Balke et al. (1993). The results exclude Euclidean 2-D diffusion but are consistent with percolation theory for diffusion of clusters at a density below the percolation threshold (Balke et al., 1993; Lawrence and Schrijver 1993).

Other methods to analyze fractals in the photospheric magnetic field in active regions focus on the scaling behavior of the structure function, applied to the longitudinal magnetic field (Abramenko et al., 2002), which can discriminate between weak and fully developed turbulence. Both SOC and intermittent turbulence (IT) appear to co-exist in the solar corona, since power-law avalanche statistics as well as multi-scaling of structure functions are observed simulaneously (Uritsky et al., 2007). Moreover, stochastic coupling between the solar photopshere and the corona indicate an intimate spatial connection (Uritsky et al., 2013).

Although this study is focused on the fractal geometry of transition region features observed with IRIS, we compare these results also with coronal values. The fractal dimension of coronal events has been measured for 10 X-class flares, 10 M-class flares, and the Bastille-Day flare (Aschwanden and Aschwanden 2008a; Aschwanden and Aschwanden 2008b). Interestingly, these datasets exhibit relatively large values of the fractal dimension, with a mean and standard deviation of D _A = 1.76 ± 0.14. They show a trend that the largest flares, especially X-class flares, exhibit the highest values of D _A ≲ 1.8–1.9 (Table 3). If we attribute flare events to the magnetic reconnection process, the observations imply that the flare plasma fills up the flare volume with a high space-filling factor, which is consistent with the chromospheric evaporation process.

Phenomena of smaller magnitude than large flares include microflares, nanoflares, coronal EUV brightenings, etc. Such small-scale variability events are found to have a mean fractal dimension of D _A = 1.56 ± 0.08 (Table 3), which is compatible with those found in M-class flares, but clearly has a lower fractal dimension than large flares, i.e., D _A = 1.76 ± 0.14 (Table 3).

The generation of magnetic structures that bubble up from the solar convection zone to the solar surface by buoyancy, observed as emerging flux phenomena in form of active regions, sunspots, and pores, can be statistically described as a random process, self-organization with (SOC) and without (SO) criticality, percolation, or a diffusion process. Random processes produce incoherent structures, in contrast to the coherent magnetic flux concentrations observed in sunspots. A self-organization (SO) process needs a driving force and a counter-acting feedback mechanism that produces ordered structures (such as the convective granulation cells; Aschwanden et al., 2018). A SOC process exhibits power law size distributions of avalanche sizes and durations. The finding of a fractal dimension of a power law size distribution in magnetic features alone is not a sufficient condition to prove or rule out any of these processes. Nevertheless, the fractal dimension yields a scaling law between areas ( A ∝ L D 2 ) or volumes ( V ∝ L D 3 ), and length scales L that quantify scale-free (fractal) processes in form of power laws and can straightforwardly be incorporated in SOC-like models.

If we compare the standard SOC parameters measured in observations (Figure 2) with the theoretically expected values from the standard SOC model (Table 2), we find that the power law slopes for event areas A agree well ( a A obs = 2.51 ± 0.21 ) versus a A theo = 2.33 (Figure 2), while the power law slopes for the radiated energy E agree within the stated uncertainties, ( a E obs = 2.03 ± 0.18 ) versus a E theo = 1.80 (Figure 2), but the power law slopes for the time duration T disagree ( a T obs = 2.65 ± 0.39 ) versus a T theo = 2.00 (Figure 2). The latter disagreement is possibly caused by the restriction of a constant minimum event lifetime (either 60 s or 110 s) that was assumed in the previous work (Vilangot Nhalil et al., 2020). The interpretation of these results implies that transition region brightenings have a similar statistics as the SOC model, at least for active regions, nanoflares, and large flares, with a typical fractal dimension of D _A ≈ 1.54 ± 0.04, but are significantly lower for photospheric granulation (D _A ≈ 1.23 ± 0.09), which implies the dominance of sparse quasi-linear flow structures in the photosphere and transition region.