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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