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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 DeVore-Coulomb 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 DeVore-Coulomb 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 DeVore-Coulomb” 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 DeVore-Coulomb case. This further confirms the gauge-dependance properties of each decomposition.

The time-integrated 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 DeVore-Coulomb and the general DeVore-Coulomb cases. On the other hand, as F_Bn, F_Vn only involve A, and in the general DeVore-Coulomb 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 gauge-invariant for the three derivations (Fig. A.1, right panel). While was negligible in both the practical DeVore and the DeVore-Coulomb 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 DeVore-Coulomb 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 gauge-invariant. 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 DeVore-Coulomb 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 DeVore-Coulomb method. The plotted quantities are the same as in Fig. 9.
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