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Table 1

Numerical tests of Thomson’s theorem.

B test ⟨    |fi(B)|    ⟩ E Sum

B DD 2 × 10-6 1.45 1.00 0.00 4 × 10-5 7 × 10-6 − 4 × 10-3 1.00
B TD 3 × 10-6 3.90 0.81 0.16 3 × 10-5 6 × 10-4 0.02 1.00
B MHD 2 × 10-5 1.94 0.94 0.06 1 × 10-6 1 × 10-4 − 1 × 10-3 1.00
B Ex1 4 × 10-3 4.21 0.79 0.38 5 × 10-4 0.29 − 0.46 1.00
B Ex2PP 9 × 10-4 1.51 0.88 0.11 2 × 10-4 0.14 − 0.12 1.00
B Ex2 2 × 10-3 0.72 2.29 0.14 3 × 10-4 0.94 − 2.38 1.00

B test,s

B DD,s 1 × 10-18 1.44 1.00 0.00 4 × 10-5 5 × 10-5 7 × 10-4 1.00
B TD,s 4 × 10-21 3.95 0.84 0.16 3 × 10-5 4 × 10-5 − 3 × 10-4 1.00
B MHD,s 3 × 10-21 1.94 0.94 0.06 1 × 10-6 1 × 10-6 − 1 × 10-5 1.00
B Ex1,s 6 × 10-18 5.98 0.43 0.57 2 × 10-4 5 × 10-3 − 6 × 10-3 1.00
B Ex2PP,s 2 × 10-17 3.15 0.42 0.58 1 × 10-4 2 × 10-3 − 3 × 10-3 1.00
B Ex2,s 8 × 10-18 0.99 0.61 0.39 2 × 10-4 1 × 10-3 − 2 × 10-3 1.00

Notes. The employed test fields, defined in Sect. 4, are named in the leftmost Col.. Second Col., ⟨    |fi|    ⟩: the divergence metric of the fields (see Eq. (C.2)). Third Col., E: energy of the test fields in units of 1032 erg. The BDD, BTD, BMHD fields (and their corresponding solenoidal fields BDD,s, BTD,s, BMHD,s) were rescaled assuming a maximum value of the photospheric vertical field equal to 300 G and a typical distance between the sunspot’s centers of (50, 50, 120) Mm, respectively. The successive five columns are the different contributions to Eq. (9), and “Sum” corresponds to their sum. All terms from “” to “Sum” are normalized by E. is the magnetic energy of the potential field Bp,s, that of the solenoidal component of the current-carrying one BJ,s, and are the contributions associated to the divergence of Bp and BJ, respectively, is a mixed potential-current carrying term (see Eq. (8) for their definitions).

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