The magnitude of the latter depends on the photospheric flux distribution and on the value of $\alpha$. The two values of $H_{corona}$ given in Table 7 correspond to the minimum and maximum values of $\alpha$ in the best-fitting range. Even though $\alpha$ stays below its resonant value (which would give unrealistically high magnetic helicity) in all the extrapolations, for the helicity computations a linearised expression in $\alpha$ was used Green et al. (2002). A computational box of the same extension centred on the AR was taken in all the cases. The same method was followed for AR 8100 and the results can be seen in Table 8.

### 6.4.3 Computation of the helicity generated by differential rotation

For the computation of magnetic helicity generated by photospheric plasma motions, Berger (1984, 1988) derived an expression for $dH_r/dt$ that depends only on observable photospheric quantities ($B_n$ and $\bar{v}$). Berger (1986) showed that the helicity generation rate can be understood as the summation of the rotation rate of all the individual elementary flux pairs weighted by their magnetic flux. Démoulin et al. (2002a) noticed that photospheric plasma motions generate two different helicity terms: the rotation of each polarity introduces ‘twist’ helicity, while the relative rotation of opposite polarity flux concentrations injects ‘writhe’ helicity. Twist and writhe helicities generated by differential rotation have opposite signs, while their magnitudes are similar, thus they partially cancel. The orientation of the bipole is important to understand the helicity injection by differential rotation as twist helicity dominates when the bipole is aligned parallel to the equator, while writhe dominates when it is perpendicular. Using SOHO/MDI data the helicity injected by differential rotation was computed (Table 7, column 5 – the most accurate values in the helicity budget). Green et al. (2002) followed the same method and computed the helicity injection by differential rotation for AR 8100. The results are seen in Table 8.

<table><caption>**Table 7:** The magnetic-helicity budget of AR 7978 is listed per rotation. An interval of helicity is given for the corona (3rd column) and for the cloud estimations with the observed number of CMEs (considering the two lengths (0.5 and 2 AU) of the twisted interplanetary flux tube. The budget is discussed in the text. All values are in units of $10^{42}$ Mx$^2$. Adapted from Démoulin et al. (2002b).</caption><thead><tr><th>No. of rot.</th><th>Date of CMP 1996</th><th>H<sub>cor.</sub></th><th>ΔH<sub>cor.</sub></th><th>ΔH<sub>diff. rot.</sub></th><th>ΔH<sub>magn. cl. corr.</sub></th><th>ΔH<sub>em.</sub></th></tr></thead><tbody><tr><td>1st</td><td>07 Jul.</td><td>—</td><td>≈ 7</td><td>0.2</td><td>[22, 88]</td><td>[29, 95]</td></tr><tr><td>2nd</td><td>03 Aug.</td><td>[5, 11]</td><td>12</td><td>3.</td><td>[10, 40]</td><td>[19, 49]</td></tr><tr><td>3rd</td><td>30 Aug.</td><td>[17, 23]</td><td>-9.5</td><td>3.</td><td>[6, 24]</td><td>[-7, 11]</td></tr><tr><td>4th</td><td>25 Sep.</td><td>[9, 12]</td><td>-5.5</td><td>1.</td><td>[10, 40]</td><td>[03, 33]</td></tr><tr><td>5th</td><td>23 Oct.</td><td>[4, 6]</td><td>(-1)</td><td>0.8</td><td>[8, 32]</td><td>[06, 30]</td></tr><tr><td>6th</td><td>19 Nov.</td><td>(4)</td><td>—</td><td>0.3</td><td>[6, 24]</td><td>[05, 23]</td></tr><tr><td>total</td><td></td><td>—</td><td></td><td>8.3</td><td>[62, 248]</td><td>[55, 241]</td></tr></tbody></table>

### 6.4.4 Computation of the helicity ejected via CMEs

Démoulin et al. (2002b) identified all the CMEs that originated from AR 7978 during its entire evolution using SOHO/LASCO & EIT, and Yohkoh/SXT observations. These numbers were
<table><caption>**Table 8:** The magnetic-helicity budget of AR 8100 is listed per rotation. The evolution of AR 8100 shows two different periods, one during which the AR helicity was negative (Nov. 2 to 5, 1997) and a second one when it was positive (from the second to the fifth rotation). Data for the first rotation are not included because of the uncertainty in the sign of the ejected helicity. An interval of helicity is given for the differential rotation (5th column) to reflect the evolution of the photospheric flux distribution owing to the flux emergence during the first three rotations and for the cloud estimations with the observed number of CMEs (considering the two lengths (0.5 and 2 AU) of the twisted interplanetary flux tube. The budget is discussed in the text. All values are in units of $10^{42}$ Mx$^2$. Adapted from Green et al. (2002).</caption><thead><tr><th>No. of rot.</th><th>Date of CMP 1997/1998</th><th>H<sub>cor.</sub></th><th>ΔH<sub>cor.</sub></th><th>ΔH<sub>diff. rot.</sub></th><th>ΔH<sub>magn. cl. corr.</sub></th></tr></thead><tbody><tr><td></td><td>2–5 Nov.</td><td>–</td><td>-33.6</td><td>[0.2, 0.8]</td><td>[-20, -80]</td></tr><tr><td rowspan="2">1st</td><td>02 Nov.</td><td>-11</td><td rowspan="2">33.5</td><td rowspan="2">[1.4, 5.1]</td><td rowspan="2">–</td></tr><tr><td>29 Nov.</td><td>22.5</td></tr><tr><td rowspan="2">2nd</td><td>27 Dec.</td><td>19.6</td><td>-2.9</td><td>[5.1, -4.6]</td><td>[5, 20]</td></tr><tr><td>23 Jan.</td><td>8.4</td><td>-11.2</td><td>[-4.6, -2.8]</td><td>[24, 96]</td></tr><tr><td rowspan="2">4th</td><td>20 Feb.</td><td>5.1</td><td>-3.3</td><td>[-2.8, -1.6]</td><td>[34, 136]</td></tr><tr><td>total 2–5</td><td>–</td><td>-2.0</td><td>-1.6</td><td>[19, 76]</td></tr></tbody></table>

then corrected for LASCO data gaps (26 + 5 CMEs). Assuming a one-to-one association between CMEs and magnetic clouds, i.e., interplanetary consequences of CMEs, which have clearly twisted magnetic structures, helicity ejected in a CME was estimated. For computing relative magnetic helicity per unit length in a typical magnetic cloud a numerically integrated form of Berger's equation (Berger, 1999) was used, and for the mean magnetic field the values $B_0(2 \times 10^{-4}$ G) and radius $R(2 \times 10^{12}$ cm) of 18 magnetic clouds from the sample in Lepping et al. (1990). For the length of the flux tube in the magnetic cloud two values were used: $L_1 = 0.5$ AU (DeVore, 2000), which yielded $H_r \approx 2 \times 10^{42}$ Mx$^2$ magnetic helicity, and $L_2 = 2$ AU (the cloud is still connected to the Sun; e.g., Richardson, 1997), which gave four times as much helicity for one average-sized magnetic cloud, i.e., CME. These mean helicity values had to be multiplied by the number of CMEs to obtain the total magnetic helicity ejected from this AR (Table 7, column 6).

To find the magnetic helicity shed via CMEs from AR 8100 the same method was applied. The AR was observed to produce 35 CMEs during its lifetime. After correcting for data gaps and when the AR was on the far side of the Sun the total CME count rises to 65.

### 6.4.5 Discussion on the helicity budget

AR 7978 was a classical bipolar AR oriented E-W, distorted only by the differential rotation. It had positive relative coronal helicity, (corresponding to the hemispheric helicity trend on the South hemisphere), and the differential rotation injected positive helicity as well throughout the studied six solar rotations (Table 7, 5th column). Localised small scale photospheric surface motions, as compared to differential rotation, should also inject magnetic helicity into the coronal field. Relative motions of the AR polarities during the two main flux emergence phases were clearly evident. After this period, there was no evidence of significant shearing motions other than the differential rotation (Démoulin et al., 2002b). When looking at the changes in coronal helicity from one rotation to the next, and comparing them to the amount of helicity generated by differential
rotation (4th and 5th columns in Table 7) it is obvious that differential rotation is an inefficient generator of helicity. This accentuates the importance of the helicity inflow and outflow through the boundaries of our coronal computational box. The total helicity budget of the active region may be written:

$$ \Delta H_{\text{emergence}} = \Delta H_{\text{corona}} - \Delta H_{\text{diff. rot.}} + N \cdot H_{\text{CME}}, $$

where $\Delta$ denotes the variation of the helicity, $N$ is the number of the CMEs and $H_{\text{CME}}$ is the mean helicity carried away per CME event. $\Delta H_{\text{emergence}}$ is given in the last column of Table 7. It is clear that in the helicity budget of AR 7978 the increase in coronal helicity during the first two rotations and the large amount of helicity carried away by CMEs during this period requires the largest input of helicity by the sub-photospheric layers. Indeed, major flux emergence episodes were observed then in the AR. As mentioned above, during the two episodes of flux emergence footpoint motions are thought to inject helicity in the coronal field. This helicity injection was not computed, since its contribution is included in the emergence term as defined above (Démoulin and Berger, 2003). Thus, the only surface shearing motion considered in our budget is the differential rotation. Later the emergence term in the helicity budget becomes smaller, but still not negligible. The total amount of helicity that should come from twisted flux emergence can be estimated to be between 55–241 × 10$^{42}$ Mx$^2$ in this simple bipolar region, that produced 31 CMEs. For comparison, during the same period, the differential rotation injected only 8.3 × 10$^{42}$ Mx$^2$, so it clearly was a minor contributor to the magnetic-helicity budget of AR 7978. Correction for the underestimation of the longitudinal field by MDI should bring a correction factor slightly over 2 for $\Delta H_{\text{corona}}$ and $\Delta H_{\text{diff. rot.}}$. Such correction would cause only a minor change in the total of the emergence term: $\Delta H_{\text{emergence}} = 51–246 × 10^{42}$ Mx$^2$. Though all the helicity values, especially the amount carried away in CMEs, have important uncertainties (for a discussion see Démoulin et al. (2002b); Green et al. (2002), the difference between the above mentioned total helicity numbers is significantly larger than any of the errors

AR 8100 represents an AR with a different magnetic evolution and can therefore be used to compare and contrast with that of AR 7978. The AR was born on the southern hemisphere with a bipolar configuration but within days of its first appearance secondary bipoles began to emerge and the region evolved to take on a large and complex magnetic configuration. Significant new flux continued to emerge into the region until its third rotation, after which time the flux in the AR began to decrease. The AR emerged with negative relative coronal magnetic helicity (the opposite sign to the hemispheric trend) but by its second rotation the helicity sign in the coronal field had reversed and become positive. We assign this change in helicity sign to the emergence of new flux that carried with it positive helicity as the helicity injection by differential rotation is insufficient by an order of magnitude, even though it is of the correct sign (i.e., positive).

Akin with AR 7978, columns 4 and 5 in Table 8 show that differential rotation in AR 8100 is an inefficient means of injecting helicity into the AR during rotations 1 to 4, and possibly also the 5th rotation. In contrast to AR 7978, AR 8100 displayed a strong relative motion of its main polarities throughout its lifetime. This had the consequence that, in addition to the shearing motions created by differential rotation, the polarities underwent a strong clockwise rotation of more than 150°. See Figure 27. Therefore, this AR represents a case study of a region where differential rotation is not the only shear flow acting on the coronal magnetic field. The rotation of the polarities and the effect of their orientation makes this AR interesting to study. The injection of helicity by differential rotation can be split into two components – twist and writhe – which inject helicity of opposite sign. The component that dominates depends on the orientation of the bipole. See Section 6.4.3. Initially, AR 8100 is orientated in the E-W direction and the twist helicity dominates injecting positive helicity. But as the AR rotates, it becomes more aligned to the N-S direction and writhe helicity dominates, which injects negative helicity. Then, the action of differential rotation serves to deplete helicity from the corona (which is of positive sign) rather than building it.

AR 8100 showed a high level of activity and produced 65 CMEs (corrected CME count) during
5 solar rotations – over twice that of AR 7978, which was followed for 6 solar rotations. The CME rate dropped in the second rotation but rose again in the third, probably due to the emergence of new magnetic flux, and the rate was then maintained throughout its lifetime.

AR 8100 is not alone in exhibiting a strong rotation, López Fuentes et al. (2003) found 15% of all bipoles showed significant rotation in their study of 300 regions so that for a small (but significant) sub-set of active regions, the helicity injection via differential rotation will not be constant with time.

The detailed helicity evolution of ARs 7978 and 8100 have been compared in Mandrini et al. (2004). The paper very nicely summarises that differential rotation is an ineffective source of coronal magnetic helicity in both of these very different ARs. The CMEs that are produced from the regions carry away more than 10 times the helicity injected by this mechanism, and in AR 8100 the differential rotation actually serves to deplete the coronal helicity content. What then is the source of the helicity shed by the CMEs? In the early rotations of both ARs flux emergence may provide the source. However, the flux emergence ceases in both ARs whilst the CME activity continues. The observational and theoretical considerations then point toward the helicity reservoir lying in the sub-surface portions of the AR magnetic field, which replenishes helicity into the coronal field either by a continued rise or via torsional Alfvén waves.

<figure><img src="image_29.png" /><figcaption>**Figure 27:** SOHO/MDI magnetic maps showing AR 8100 during rotations two to five. The main polarities of the active region undergo a strong clockwise rotation. Image reproduced with permission from Green et al. (2002), copyright by Springer.</figcaption></figure>

## 6.5 Evolution of coronal parameters: T, EM, ρ

The evolution of the fluxes observed in X-rays were correlated with the magnetic flux density in active region NOAA 7978 from its birth throughout its decay. Magnetic observables were derived from SOHO/MDI data while Yohkoh/SXT and Yohkoh/BCS data were used to determine the global evolution of the temperature and the emission measure of the coronal plasma at times when no significant brightenings were observed. It was shown that the mean X-ray flux and derived
parameters, temperature and emission measure (together with other quantities deduced from them, such as the density and the pressure), of the plasma in the AR follow power-law relationships with the mean magnetic flux density ($\bar{B}$) (Figure 28). The exponents ($b$) of these power-law functions ($a\bar{B}^b$) were derived using two different statistical methods, a classical least-squares method in log-log plots and a non-parametric method, which takes into account the fact that errors in the data may not be normally distributed. Both methods give similar exponents, within error bars, for the mean temperature and for both instruments (SXT and BCS); in particular, $b$ stays in the range [0.27, 0.31] and [0.24, 0.55] for SXT images and BCS data, respectively. For the emission measure the exponent $b$ lies in the range [0.85, 1.35] and [0.45, 1.96] for SXT and BCS, respectively (see Figure 29.

<figure><img src="image_30.png" /><figcaption>**Figure 28:** Log-log curve of the flux per unit surface area observed with the Al0.1 and AlMgMn filters temperature and emission measure derived from Yohkoh/SXT versus magnetic flux density ($\bar{B}$). The solid line shows the linear least-squares fit and the two dotted lines correspond to the $3\sigma$ error in the slope of the solid curve. Only the decaying phase (points mark with ◊) are included in the least-squares fit. The same analysis has been done for Yohkoh/BCS data. Image reproduced with permission from van Driel-Gesztelyi et al. (2003b), copyright by AAS; see also van Driel-Gesztelyi et al. (2003a).</figcaption></figure>

As a next step, Démoulin et al. (2003) derived the dependence of the mean coronal heating rate on the magnetic flux density (Figure 29). They used the scaling laws of coronal loops in thermal equilibrium to derive four observational estimates of the scaling of the coronal heating with $\bar{B}$ (two from SXT and two from BCS observations). These results are used to test the validity of coronal heating models. It was found that models based on the dissipation of stressed, current-carrying magnetic fields are in better agreement with the observations than models that attribute coronal heating to the dissipation of MHD waves injected at the base of the corona. This confirms, with
smaller error bars, previous results obtained for individual coronal loops (Mandrini et al., 2000) as well as for the global coronal emission of the Sun and cool stars (Schrijver and Aschwanden, 2002). Taking into account that the photospheric field is concentrated in thin magnetic flux tubes, both SXT and BCS data were found to be in best agreement with models invoking a stochastic buildup of energy, current layers and MHD turbulence.

<figure><img src="image_31.png" /><figcaption>**Figure 29:** Ranges for the exponents found for the mean physical parameters in the scaling law equations: parameter $\propto B^{\text{exponent}}$ ($\bar{B}$: magnetic flux density). The temperature, T, and the emission measure, EM, are derived directly from the observations, then the plasma density, $N_e$, and pressure, P, are deduced. $T_{\text{max}}$ is the maximum temperature derived from the quasi-static thermal model of coronal loops. $H_T$ and $H_P$ are the heating rate derived from the observations and the thermal model. For $T_{\text{max}}$ and $H_P$ two values of $\alpha$ are shown (coronal radiative losses $\propto T^{\alpha}$). The $\pm 3\sigma$ error range of a normally distributed statistics is light shaded. The 90% confidence interval of the non-parametric statistics is dark shaded. Image reproduced with permission from Démoulin et al. (2003), copyright by AAS.</figcaption></figure>

## 6.6 Evolution of multi-wavelength emitted fluxes

The long-term evolution of AR 7978 led to rotational modulation of the emitted fluxes with an amplitude depending on the observing wavelength range, as shown in Section 6.2.1. The evolution of the multi-wavelength emitted fluxes/ flux densities from optical to X-rays as a function of the evolving magnetic field carry information about heating mechanisms acting at different atmospheric levels. Such a quantitative analysis was carried out, taking one set of data per rotation at each consecutive central meridian passage of AR 7978, outside the time of flares, obtained from SOHO (MDI, EIT, CDS, SUMER and CELIAS/SEM), Yohkoh (BCS, SXT), GOES, SOLSTICE and 10.7-cm radio data from DRAO, Canada (for a subset, see Figure 30). Preliminary results of the analysis were published in van Driel-Gesztelyi et al. (1999b) and van Driel-Gesztelyi et al. (2001), and a summary of the final results by Démoulin (2004).

The 19 emitted-flux datasets were obtained with different types of instruments: imager with filters (EIT), imager with spectrograph (CDS, SUMER), Sun-as-a-star spectrometer (SOLSTICE, CELIAS) and radio telescope (DRAO). From all these data, the mean radiative excess flux densities were derived $\Delta F = (\text{flux} - \text{basal flux})$ per unit surface area for each rotation for each dataset (basal
<figure><img src="image_32.png" /></figure>

**Figure 30:** Multiwavelength images of AR 7978 during four rotations (2nd–5th) showing the evolution of magnetic fields (SOHO/MDI data, upper left), 304 Å He II (SOHO/EIT, upper right), 171 Å Fe IX (SOHO/EIT, lower left), and SXRs (Yohkoh/SXT, lower right). Note that the spatial distribution of the emission in hotter lines (1–2 MK) is similar to that in 304 Å, but as the magnetic field was getting dispersed, and the loops grew, there was a re-organisation of the emission with height. At earlier times hot and dense plasma fully filled the loops, while later the EUV loops were only partially filled, as they became taller than the gravitational scale height. In SXRs ($T \approx 2–5$ MK) more fully filled loops were observed besides an increasing presence of a diffuse emission.
flux is attributed to heating of non-magnetic origin, e.g., acoustic wave flux). Then the dependence of the 19 $\Delta F$ values on the photospheric mean magnetic flux density $\bar{B}$ was analysed. Démoulin (2004) found power-law relations between the detected excess flux density and the magnetic flux density in these different spectral domains, $\Delta F \propto \bar{B}^b$, with the exponent b depending on the formation temperature of the spectral lines involved. For the SXRs the dependence was found to be almost quadratic ($b = 1.6-2.6$), while for the radiation emitted from the transition region and the chromosphere the dependence of the measured fluxes was found to be more linear ($b = 0.7-1$). However, Démoulin (2004) proposed that the measured SXR fluxes are strongly affected by the instrumental response function $R(T)$, which increases with $T$ instead of peaking at a given $T$. The power-laws illustrated in Figure 31 were derived from data corrected for this instrumental effect using the observed mean $T$. In this corrected dataset the power-law exponents for various SXR data were found to be in the range $b = 1.1-1.2$, closer to the exponents found for transition-region and chromospheric emission.

<figure><img src="image_33.png" /><figcaption>**Figure 31:** Ranges for the exponents, b, found for the power-law relations between the detected excess flux density and the magnetic flux density in different spectral domains, $\Delta F \propto \bar{B}^b$. The results are approximately ordered along the abscissa according to the mean temperature of the emitting plasma (except when the ranges overlap, e.g., in X-rays). Results at the extreme right, outside of the box, have such a broad wavelength (temperature) range that a mean temperature cannot be defined. The ordinate extension of the shaded regions represent the $\pm 3\sigma$ error range. The Sun-as-a-star measurements are dark shaded, while the spatially resolved measurements are light shaded, with the lightest shading for the spectrometers (CDS, SUMER). The horizontal grey band corresponds to the range of the exponent for the coronal plasma pressure. In the X-ray range, the exponents are corrected for the temperature dependence of the instrumental response function. Image reproduced with permission from Démoulin (2004), copyright by IAU.</figcaption></figure>

What is the relationship between the power laws found in this study and the power law dependence between total X-ray luminosity ($L_x$, total flux) of ARs and total magnetic flux ($\Phi_{tot}$): $L_x \propto \Phi_{tot}^{1.19\pm0.04}$ (Fisher et al., 1998) and for a wide range of features from X-ray bright points to red dwarfs and T Tauri stars covering over 12 magnitudes in magnetic flux: $L_x \propto \Phi_{tot}^{1.15\pm0.05}$ found by Pevtsov et al. (2003)? Obviously, the power-law exponents are very close to those found by Démoulin (2004) for X-rays, after a correction for the instrumental response functions. However, Démoulin (2004) argues that there is a natural correlation that exists between $L_x$ and $\Phi_{tot}$ (extensive quantities) through the spatial extent of the magnetic structures involved. To make sure that the approximately linear relationship between SXR flux and magnetic field is not due to a natural proportionality of the extensive quantities with the magnetic area considered, which may
dominate over the relatively smaller variations of the intensive quantities, the intrinsic physical relations between intensive quantities ($\Delta F$ and $\bar{B}$), i.e., flux densities should be analysed. In the study by Fisher et al. (1998) SXT DN was transformed into flux values by assuming a fixed $T$ of 3 MK, therefore the $T$ dependence of the instrumental response function was not taken into account. This may not have been important for that study, however, as the analysed ARs were approximately in the same evolutionary stage, i.e., their $T$ was probably similar. In the analysis by Pevtsov et al. (2003) the $T$ was estimated for most of the different magnetic structures accounting for the $T$ dependence of the instrumental response function. As $R(T)$ appears to be the underlying cause of discrepancy between exponents found between X-ray fluxes/flux densities and magnetic variables, accounting for this brings the relationship close to being linear both on the Sun and cool stars (see references in Démoulin, 2004).

To interpret the transition region (TR) flux densities' relationship with $\bar{B}$, Démoulin (2004) concluded that the four mechanisms previously proposed for heat transport in the TR result in scaling laws, which are compatible with those deduced from the observations shown in Figure 31. The four mechanisms are: (i) classical thermal conduction by electrons (Spitzer, 1962; Lie-Svendsen et al., 1999), (ii) enhanced conduction by turbulence (Cally, 1990), (iii) ambipolar diffusion of ions and neutral atoms (Fontenla et al., 1990), and (iv) transport by plasma flows (Chae et al., 1997).

Interpreting the power laws found for emission from the chromosphere, the best model to be compatible with the power-law exponents found was that by Solanki et al. (1991) (see Figure 32). However, the emission from the chromosphere is not simply linked to the energy deposited in the region where an emission line is formed, as radiation from the photosphere and neighbouring chromospheric layers also contribute.

<figure><img src="image_34.png" /></figure>

**Figure 32:** Schematic representation of models of the emission of chromospheric lines outlining the main effects of a changing magnetic filling factor (longterm evolution from right to left). The emitting layer is represented by the darker gray bands. Observations are not able to resolve the flux tubes, but measure the mean excess flux, $\Delta \bar{F}$, and the mean magnetic flux, $\bar{B}$. The same atmosphere is present as $\bar{B}$ decreases from right to left, while the merging height increases. This leads to a natural decrease of $\Delta \bar{F}$ with decreasing $\bar{B}$. Owing to the changing merging height with $\bar{B}$, $\Delta \bar{F}$ saturates for increasing $\bar{B}$ as the magnetic flux tubes get into contact in the emission region (on the right). This is a representation of the hydromagnetic model by Solanki et al. (1991) of an expanding flux tube for the interpretation of Ca II K line flux, and explains well the results for the evolution of chromospheric emission during the evolution of AR 7978 by Démoulin (2004), from where the image is reproduced with permission, copyright by IAU.

## 6.7 What did we learn from long-term evolution studies of ARs?

The results presented above on the long-term evolution of ARs confirmed that the evolution of an active region in each and every detail reflects the evolution of its magnetic field. It was found that flares mainly occur when the magnetic field of the AR has the highest complexity and magnetic flux density, and this is when the variability of AR cores are the highest, too. However, CMEs may be more closely related to the magnetic helicity content of the AR. It was also studied how much the contribution of differential rotation and twisted flux emergence contribute to building up the observed shear and helicity. Differential rotation turned out to be a very ineffective generator of
magnetic helicity, thus the dominant source of coronal and interplanetary helicity is the inherent twist of the flux tubes forming the ARs. Magnetic helicity is transferred from the sub-photospheric layers to the corona by magnetic flux emergence and replenished after relaxation (CME) events either by a slow continuous emergence of the flux tube or by torsional Alfvén waves (Longcope and Welsch, 2000). It being concentrated along the magnetic inversion line by magnetic cancellations there. This process can continue for several solar rotations until the helicity reserves of the flux tube are exhausted, or the flux tube is destroyed by convective motions.

A study of how atmospheric heating is affected by the magnetic evolution was carried out by van Driel-Gesztelyi et al. (2003b) and Démoulin et al. (2003). Using Yohkoh/SXT and BCS data they studied the evolution of the coronal plasma parameters as functions of magnetic observables during five solar rotations and derived scaling laws between them (van Driel-Gesztelyi et al., 2003b). It was found that the mean X-ray flux and derived parameters, temperature and emission measure of the plasma (together with other quantities deduced from them, such as the density and the pressure), in the AR follow power-law relationships with the mean magnetic flux density. The determination of such power-law relations, when combined with scaling laws deduced from coronal heating models, can provide us with powerful tools for determining the mechanisms responsible for the existence of the high temperature corona (Démoulin et al., 2003). Through such a study it was found that models involving the dissipation of stressed, current-carrying magnetic fields as well as magnetic turbulence are in much better agreement with the scaling laws derived from the long-term evolution of AR 7978 than the ones which solely involve wave heating.

Another complex analysis involved the use of multi-instrument datasets leading to a study of the evolution of emitted flux densities in 19 different wavelength domains in correlation with magnetic flux density. Analysing the obtained power-laws and comparing them to heating mechanisms and models of different layers of the solar atmosphere, Démoulin (2004) concluded:

> We have learned that the magnetic field is not only a strong structuring agent of the solar atmosphere, but also that it defines its main properties (such as pressure and temperature). There is plausibly a unique heating mechanism in the magnetised part that heats both the upper chromosphere and the corona [...]. The heating rate scales approximately with the magnetic field to the second power (transforming all the parameter dependence to the magnetic one). In the intermediate layer, the TR, this heating mechanism could well be at work, but the presence of such high radiative losses makes this heating input unimportant for the TR physics. Finally, taking into account the observed universality of the flux-flux relations (power laws between fluxes from the optical to the X-ray range), this physical scenario is expected to extend to cool stars.

The latter was one of the results relevant to stellar activity studies. However, emitted flux evolution leading to evolving rotational modulation was also used to successfully test photometric starspot modelling programs (Oláh et al., 1999) and interpret stellar X-ray spectra (Orlando et al., 2004). The long-term evolution of ARs, especially during solar-minimum conditions, is an outstanding tool for creative solar and stellar physicists to decipher components of complex phenomena covered by their observations.
# 7 Concluding Remarks

We started this review with references to IAU Symposium 35 on the “Structure and Development of Solar Active Regions”, held in Budapest, Hungary in September 1967, and we would like to end with another reference to it. The usual welcome speech at the start of the Symposium was given by László Detre, who was then the director of Konkoly Observatory and a famous expert on variable stars, who cited a few lines from a poem published in the latest issue of the *Irish Astronomical Journal* which arrived to Konkoly Observatory’s library prior to the Symposium. The title of the poem was “A Solar Physicist’s Lament” and, we believe, these lines have not lost their validity (nor their charm) in nearly 50 years:

> “It looks as if many a sunny day
> Will pass along on its way
> Before we solve it all.”

As active region evolution has relevance to most aspects of solar activity, more and even deeper analyses of AR evolution as well as studies done in an AR evolutionary context will help us to “solve it all”.

# Acknowledgements

The authors thank Deb Baker, Pascal Démoulin and two anonymous referees for a critical reading of the manuscript and insightful comments. LvDG’s work was supported by the STFC Consolidated Grant ST/H00260/1 and the Hungarian Research grant OTKA K-109276. LMG is grateful to the Royal Society for a University Research Fellowship.
# A Appendix: Sunspot Classification Schemes

Throughout the centuries of sunspot observations there have been many attempts to classify sunspot groups. Though these classifications may appear like “zoology”, the “Zürich classification” scheme devised by Waldmeier (1955) and further developed by McIntosh (1990), is worth describing here, since it is based on the characteristic stages that a spot group may pass through during its formation and decay. There are many varieties of this scheme. Here we largely follow the scheme as summarised in the User guide from the Solar Influences Data Analysis Center (SIDC) of the Royal Observatory of Belgium.

The 3-component McIntosh classification is based on the general form ‘Zpc’, where ‘Z’ is the modified Zürich Class, ‘p’ describes the penumbra of the principal spot, and ‘c’ describes the distribution of spots in the interior of the group.

Z-values: (Modified Zürich Sunspot Classification).

**A** - A small single unipolar sunspot. Representing either the formative or final stage of evolution.

**B** - Bipolar sunspot group with no penumbra on any of the spots.

**C** - A bipolar sunspot group. One sunspot must have penumbra.

**D** - A bipolar sunspot group with penumbra on both ends of the group. Longitudinal extent does not exceed 10°.

**E** - A bipolar sunspot group with penumbra on both ends. Longitudinal extent exceeds 10° but not 15°.

**F** - An elongated bipolar sunspot group with penumbra on both ends. Longitudinal extent of penumbra exceeds 15°.

**H** - A unipolar sunspot group with penumbra.

p-values:

**x** - No penumbra (group class is A or B)

**r** - Rudimentary penumbra partially surrounds the largest spot. This penumbra is incomplete, granular rather than filamentary, brighter than mature penumbra, and extends as little as 3" from the spot umbra. Rudimentary penumbra may be either in a stage of formation or dissolution.

**s** - Small, symmetric (like Zürich class J). Largest spot has mature, dark, filamentary penumbra of circular or elliptical shape with little irregularity to the border. The north-south diameter across the penumbra is ≤ 2.5°.

**a** - Small, asymmetric. Penumbra of the largest spot is irregular in outline and the multiple umbra within it are separated. The north-south diameter across the penumbra is ≤ 2.5°.

**h** - Large, symmetric (like Zürich class H). Same structure as type ‘s’, but north-south diameter of penumbra is ≥ 2.5°. Area, therefore, must be ≥ 250 MSH (millionths of the solar hemisphere).

**k** - Large, asymmetric. Same structure as type ‘a’, but north-south diameter of penumbra is more than ≥ 2.5°. Area, therefore, must be ≥ 250 MSH.
c-values:

x - Undefined for unipolar groups (class A and H)

o - Open. Few, if any, spots between leader and follower. Interior spots of very small size. Class E and F groups of 'open' category are equivalent to Zürich class G.

i - Intermediate. Numerous spots lie between the leading and following portions of the group, but none of them possess mature penumbra.

c - Compact. The area between the leading and the following ends of the spot group is populated with many strong spots, with at least one interior spot possessing mature penumbra. The extreme case of compact distribution has the entire spot group enveloped in one continuous penumbral area.

Another useful classification scheme, still widely used in the literature, devised by Hale and his colleagues at Mt. Wilson Observatory (Hale et al., 1919), refers to the magnetic complexity of sunspot groups. The scheme was widened to include the most complex class: δ-groups (Künzel, 1960).

α - Denotes a unipolar sunspot group.

β - A sunspot group having both positive and negative magnetic polarities, with a simple and distinct division between the polarities.

β – γ - A sunspot group that is bipolar but in which no continuous line can be drawn separating spots of opposite polarities.

γ - A complex active region in which the positive and negative polarities are so irregularly distributed as to prevent classification as a bipolar group.

δ - A complex magnetic configuration of a solar sunspot group consisting of opposite polarity umbrae within the same penumbra.
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