tokens: of the field defined by the generating cylinder, which was used for the construction of the flux rope shown in panel a) , i.e., the field ( 9 )–( 11 ). In contrast to panel a) , they are shown in the whole field of view because the magnetic field from all space contributes to the solution displayed in panel a) . The cross section of the generating cylinder is plotted by the dashed circle, and the tilde coordinates are the local coordinates defined in Fig. 1 . The magnetic-field magnitude scale is the same, but the values of are different and correspond to the maxima in the displayed fields in panels a) and b) . The thick lines indicate magnetic surfaces on which the axial magnetic field is zero. This method is able to generate toroidal flux ropes with another cross sections. Figure 6 displays shapes very close to ellipses, a prolate shape in Fig. 6 a and an oblate one in Fig. 6 b. Only ten coefficients were used to define these fields (it was , and half of the coefficients were close to zero, as in the previous case). The axial field does not vanish at the boundary shown, but it is small, as is shown in Fig. 3 . There are limitations of our procedure. One has been mentioned: it was not commonly possible to obtain the desired shape of the flux rope for low aspect ratios. But even if the desired shape is obtained, the flux rope may be multipolar, that is, with the axial field reversed inside. For instance, the generating toroid of the flux rope shown in Fig. 6 a has an oblateness 2:3, but the flux rope has a dual polarity. Therefore we limited it to the inner unipolar part, displayed in Fig. 6 a, which has a smaller oblateness and a different aspect ratio (if defined as ) from 5. If the generating toroid has an oblateness or prolateness larger than about two, unipolar flux-rope cross sections deviate from desired elliptic shapes (they more resemble a rectangular shape). Fig. 6 Magnetic field magnitude distribution inside toroidal flux ropes with practically elliptic cross sections. The closed lines are cross sections of magnetic surfaces. The values of free parameters are a) , , , , , , , , , , , and ; b) , , , , , , , , , , , and . The other coefficients are (numerical) zero. 3.1. Calculating the helicity It is relatively easy to (numerically) calculate the helicities of these flux ropes. Introducing toroidally curved cylindrical coordinates , , and in our global system, the helicity for toroidal flux ropes with an axially symmetric linear force-free field ( Vandas Romashets 2015 ) is where determines the boundary; and are polar coordinates in a cross-section plane (like in Figs. 5 and 6 , where we should have for the case of Fig. 5 a and for Fig. 6 , where and are semi-axes of the boundary ellipses; for a more general boundary, would be defined numerically from the results). To compare different flux ropes, Vandas Romashets (2015) used the unitless quantity derived from the helicity by where is the volume of the toroidal flux rope and
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