We can parameterize the torus T with the coordinates (θ,z) where 0≤θ<2π and z∈C, |z|≤1. Let a be an integer, a≥1, b∈C, |b|<1, c∈C, and d an integer, d≥1 (unless a=1 in which case d≥0) and gcd(a,d)=1 (Fig. 3 to the left).

Consider the map F:(θ,z)↦(aθ,bz+ceidθ). Hence F maps the torus into a torus that has been “folded” a times.

We assume that b and c are chosen so that F maps the torus into itself. Consider the set F(T) at the section θ=θ0. The parameter a is the number of connected components of F(T).

The preimages of the points in F(T) at the section θ=θ0 are the points with θ=θ0a+2πak,k=0,1,…,a-1.

Hence, the section of F(T) at θ=θ0 is the set {bz+ceid(θ0/a+2πk/a):|z|<1,k=0,1,…,a-1}.

We can therefore get some information about the parameters according to the picture to the right in Fig. . For k=0,1,…,a-1 we have ϕk=γ+d(θ0a+2πak), where ϕk is defined in Fig.  and γ is such that c=|c|eiγ.

We have now seen that we can extract complexity parameters from a picture.

Let us now address the second question and try to reproduce magnetograms from different values of these complexity values for the torus.

With b=|b|eiδ we can write the map F as F(θ,z)=(aθ,|b|eiδz+|c|eiγ+idθ).

By changing the values of a, b, c, d and the angle θ we can describe by Fn(T), where the number of iterations n is a positive integer, a linked, a twisted and writhed solid torus. The parameter a describes how many times the curve winds about the centre, |b| changes the thickness of the image of the solid torus, |c| determines the separation of the solid torus parts in the cross-sectional planes, (a-1)d is the linking number of the image of the solid torus and finally the parameter δ determines the twist of the torus. The solid torus is then cut along its circular axis in two parts. These are the blue and red parts shown in Fig. . The two parts will be treated as positive and negative poles.

In Fig.  we give a couple of examples. The first solid torus to the left shows the original torus. By changing the parameter d to 1 or 3 we obtain a one- or a three-linked torus. By changing δ from 0 to 2 for the original torus we obtain a twisted torus. Finally by changing the c to 1 and d to 3 for the original torus we obtain a writhed torus.

We cut the torus at an angle θ and then calculate a simulated magnetogram from this cut as follows: for each point y in the simulated magnetogram, the intensity is given by the integral ∫ρ(x)|y-x|-2dV(x), where the integral is over all point x in the cut, and dV denotes the area measure. The function ρ(x) is defined to be 0 if x is outside the torus, and ρ(x)=±1 depending on which part of the torus x is in. In the computer, this integral is approximated by a finite sum.

We can also simulate magnetograms at any time in between. The colour code has been chosen to be the same as for observed HMI, SDO magnetograms. The most simple simulated magnetogram is obtained by taking a cut of the original torus into two adjacent tori with halved cross-sectional area (treated as positive and negative poles) and a grid of values is calculated based on their inverse distance squared. We may then e.g. map the torus once (n=1), and take cuts at θ = 120∘ or θ = 360∘ and obtain the other magnetograms (Fig. ).

The solenoid is an attractor which is contained in a “solid torus” . We would therefore expect iterating F should produce fractal magnetograms.

The magnetograms in Fig.  show exactly that.

Before trying to reproduce the magnetograms of May in 1921 we give two examples of more recent magnetograms observed by HMI on SDO (Fig. ). As can be seen we capture the general structure, but a more fractal structure should have been included. For more complicated regions we also need several connected tori. In the next and final example of May 1921 we use four connected tori.

Finally, we address the third question. We will reproduce the magnetograms of May 1921 and then try to understand what caused the extreme solar storm and when it occurred.

We use four tori to reproduce the magnetograms for 12 to 16 May (Fig. ). It is assumed that these four tori are connected. The parameter sets used are displayed in Table 1.

We start with 12 May at 18:00 UT and after the break-up into two major regions which seems to have taken place on 13 May UT morning. We then try to reproduce the changes in the magnetograms topologically, i.e. by continuous change of θ and z. On 13 May the left region starts to rotate anticlockwise to follow the Hale law. The right region starts to rotate clockwise. On late 13 to early 14 May we notice both emerging of flux and large rotations especially for the active region right-hand side. A dramatic change seems to take place. We were unable to reproduce the observed magnetograms by continuous changes of θ and z, but had to rotate the simulated magnetogram. We therefore find it probable that this rotation, caused by the opposite rotation of the two main pairs of opposite polarity, produced a topological change and that reconnection had taken place which would have explained the energy release and thus the extreme solar storm of 14 May. The CME then reached Earth at about 22:00 UT on 14 May.

There are many possible extensions of the solid torus approach. A natural extension is to start trying to reconstruct magnetograms of higher time and spatial resolution, such as the magnetograms produced by HMI onboard SDO , something we have already prepared for. With measurements of vector magnetic fields we will be able to make estimation of the energy release and give a better description of the magnetic complexity.

It would be very interesting to try to estimate the energy released during the extreme solar storm based on the change of the complexity parameters of the torus. estimated that the free energy Em, stored in the braided field, is proportional to the square of the crossing number Cmin: Em≥9.06×10-2Cmin2Φ2N2L, where Φ is the magnetic flux of the flux tubes, N the number of strands of the braid (= flux tubes) and L the length. The linking numbers are closely related to the average crossing number, which is an algebraic measure of the link complexity in space . Interestingly it is also found that the energy released due to reconnection of the braids in the coronal loops follows a power-law distribution, i.e. is fractal.

We have used several solid tori to describe large complex active regions and the evolution. It would also be interesting to study the small–large-scale magnetic field coupling as seen at times of solar flares and the Hale Solar Sector Boundary .

How quickly might a severe solar storm develop into an extreme storm? With an estimate of the energy release based on the parameters of the torus, this would be an interesting issue to examine. In the case of the 1921 event it took less than a week. During the Halloween events in 2003 it also took less than a week between the severe solar storms of 28 and 29 October and the extreme solar storm of 4 November. During the most recent event in July 2012 it took more than a week when the active region was on the far side . The solar storms of AR 11 520 in July 2012 reached a β–γ–δ. For that occasion we can also use the parameters describing the complexity based on SDO observations and available complexity parameters through Space weather Helioseismic and Magnetic Imager Active Region Patches (SHARP) , something which will be further discussed in an upcoming paper. The region grew to a size of 1460 millionths, an intrusion of negative polarity flux occurred in the positive umbra of the spot on 12 July at 13:00 UT and disappeared on 14 July at 09:00 UT. It produced an X 1.4 solar flare. A halo CME also occurred producing a proton event of about 100 pfu. An interesting coronal S-shaped sigmoid structure occurred just before the onset of the solar flare. One may therefore suspect that a kink instability occurred . Not until it was on the far side did the active region 11 520 become an extreme solar storm. On 23 July it produced a very fast CME of 3400 km s-1 . We therefore expect that an extreme solar storm occurred on the far side of the Sun. Based on observations by STEREO of the velocity and magnetic field, a model was used to calculate a hypothetical dB/dt if the CME was headed toward Earth. A value somewhat larger than 1000 nT min-1 was found, making it a candidate for an extreme solar storm. The geomagnetic storm index for the 2012 event was estimated as -1154 nT, larger than that of the Carrington event at -850 nT .

Finally, a follow-up of the work in would be to examine whether or not the seeming lack of coupling between the intensity of the extreme solar storms and intensity of the cycle has been the case only for recent cycles.