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Appendix A: Helicity test without the condition A (z_{top}) = A (z_{top})
In the numerical implementation of both the practical DeVore method and the DeVoreCoulomb methods, we imposed the condition of Eq. (46), of having the same distribution of A and A_{p} at the top boundary, in addition to the DeVore gauge (Eq. (45)), which is valid in the whole volume. Because of this condition, A and A_{p} are both different when computed with one method or the other. The gauge of A and the gauge of A_{p} are linked by Eq. (46). It induces that F_{AAp} is null at the top boundary and reduces its intensity in the whole domain.
It is possible, however, to compute and estimate the helicity without enforcing Eq. (46). We can follow the helicity evolution by mixing the vector potential computed with both methods. Since the gauge invariance of Eq. (10) does not require using the same gauge for A and A_{p}, we can use the A computed with the practical DeVore method with A_{p} derived with the DeVoreCoulomb methods. A_{p} thus still satisfies both the DeVore and the Coulomb gauge. As A, which is only satisfying the DeVore gauge condition, has been computed independently of A_{p}, there is no boundary surface along which they share any common distribution. In this appendix we refer to this derivation as the “general DeVoreCoulomb” case.
Figure A.1, right panel, presents the different terms entering in the decompositions Eqs. (11), (15) of the relative helicity. As in Sect. 5.1, the gauge invariance of in this computation relative to the other methods is ensured with a high precision (<0.3%). As with the others methods, H_{j} and H_{pj} remain constant, while H_{m}, H_{mix} and H_{p} are different in the general DeVoreCoulomb case. This further confirms the gaugedependance properties of each decomposition.
The timeintegrated helicity fluxes (Fig. A.1, right panel) again show that tightly follows the variation of helicity
. The helicity dissipation is also very small in the other two methods and has a precision similar to the practical DeVore case (Sect. 4.3). The repartition of the helicity flux F_{tot} between the different terms that compose it is significantly different here compared to the other two methods, however.
Since A_{p} fulfills the Coulomb condition, the term dH/ dt_{p,var} is null to the numerical precision. Since F_{φ} only involves quantities based on the derivation of the potential field, are equal for both the DeVoreCoulomb and the general DeVoreCoulomb cases. On the other hand, as F_{Bn}, F_{Vn} only involve A, and in the general DeVoreCoulomb are equal with their respective estimations in the practical DeVore case.
F_{AAp} concentrates the helicity flux contribution, which enables F_{tot} to be quasi gaugeinvariant for the three derivations (Fig. A.1, right panel). While was negligible in both the practical DeVore and the DeVoreCoulomb cases, we observe that this term is now a main contributor of the helicity fluxes. This is expected since results from the existence of large differences between the distribution of A and A_{p} on the boundaries. The computations in the practical DeVore and the DeVoreCoulomb method both enforced Eq. (46), which induces a very weak value of . We observe that dropping condition (46) creates a strong .
This test again demonstrates that the choice of the gauge strongly influences the distribution of the helicity fluxes that compose the total helicity flux F_{tot}. Only the total flux F_{tot} is quasi gaugeinvariant. None of the terms that compose the helicity flux F_{tot} must be neglected a priori. Depending on the gauge, each term can carry a significant contribution. In a numerical estimation, it is thus highly advisable to compute all the terms that form the helicity flux density (Eq. (23)). Explicitly computing each term allows us to verify that the constraints set on the used gauges are effectively enforced numerically.
Fig. A.1
Left panel: relative magnetic helicity (, black dashed line) and its decomposition in the general DeVoreCoulomb method. The plotted quantities are the same as in Fig. 2, bottom panel, and Fig. 7, right panels. Right panel: and H_{∂𝒱,#} evolution in the system computed with the general DeVoreCoulomb method. The plotted quantities are the same as in Fig. 9. 

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