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Appendix A: Details of the numerical implementation

In the applications presented in this paper we have considered uniform Cartesian grids of resolution Δ in all directions, discretizing a rectangular volume (see Sect. 4 for the actual values of Δ and in each case). We compute derivatives using the standard second-order, central-difference operator, and we employ the relevant one-sided (i.e., forward or backward), second-order differences at the boundaries of . The only exception is the computation of the divergence of B_test, since all test fields are known in a volume that is larger than the selected (on lateral and top boundaries). In this case, ∇·B_test is computed using the central differences also at the location of the lateral and top boundaries of .

In the computation of volume integrals, the cell volume Δ³ is assigned to each internal node of the grid, whereas the cell volume is reduce to half, one fourth, and one eighth for nodes on the lateral surfaces, edges, and corners of , respectively. Similarly, in the computation of surface integrals, the cell surface Δ² is assigned to each node inside each side of , whereas the cell surface is reduced to half and one fourth on edges and corners of each side, respectively. Despite the accurate computation of integrals, the divergence theorem, Eq. (4), is not insured to hold numerically, a property that requires special techniques, like finite-volume discretizations, to be fulfilled.

Appendix B: Divergence cleaner

To construct a numerically solenoidal field [B_s] from a field [B] let us define (B.1)where A is the vector potential computed from B in the volume . The vector potential A can be derived as in Valori et al. (2012) using the gauge ẑ·A = 0, yielding the expression (B.2)where b ≡ (A_x(x,y,z = z₂),A_y(x,y,z = z₂),0) is any solution of (B.3)A direct substitution of Eq. (B.2) into Eq. (B.1) shows that (B.4)with the property that ∇·B_s = 0. In other words, Eq. (B.4) naturally separates B into a solenoidal part B_s and a nonsolenoidal one, thus defining a divergence cleaner for B. The z-component of B is changed throughout the volume except on the top boundary, whereas the x- and y-components are unchanged. The amplitude of the modification to B at a given height z is given by the cumulative effect of “magnetic charges” above that altitude.

Since only the z-component of the field is changed, the divergence cleaner changes the x- and y-components of the current, but not the z-component, (B.5)

therefore the cleaner changes the injected magnetic flux but not the injected electric current through the bottom layer. On the other hand, since most of the test fields considered in this article have the highest values of divergence close to the bottom boundary, only the lower part of the field is changed significantly by the cleaner.

Computation B_s requires numerical computation of an integral of the type , as in Eq. (B.4) for f = ∇·B. To achieve numerical accuracy in the solenoidal property of B_s, G(z) must satisfy ∂_zG(z) = −f(z) numerically, i.e., must satisfy the numerical formulation of the fundamental theorem of integral calculus in the employed discretization. For the second-order central differences that are used in the analysis, this can be obtained by the recurrence formulae (B.6)where G(z) = G(z₁ + kΔ) ≡ G(k) with k = 0,1,2,···   ,(n_z − 1), and Δ is the uniform spatial resolution in z.

The constraint ∂_zG(z) = −f(z) in the second-order, central-difference discretization does not fix the value of G(n_z − 2). To do that, we require that the divergence of Eq. (B.4) also vanishes at the bottom boundary, i.e., (∇·B_s)|_z = z1 = 0. Here the second-order divergence operator is computed by using a second-order, forward derivative in the z-direction, i.e., defining the operator , where and . By using the recurrence formula Eq. (B.6), the condition on the bottom boundary is transformed into the condition for G(n_z − 2), yielding where f_os = ∇_os·B. Such a numerical trick is only possible if the volume is discretized by an even number of points in the z-direction, therefore the analysis volumes employed in the article were chosen to satisfy such a requirement.

Appendix C: Measures of ∇ · B

The total divergence of a field B can be conveniently expressed by a single number using the average ⟨    |f_i|    ⟩ over the grid nodes of the fractional flux (C.1)through the surface ∂v of a small volume v including the node i (Wheatland et al. 2000). Taking a cubic voxel of side equal to Δ as the small volume v centered on each node, the divergence in the discretized volume of uniform and homogeneous resolution Δ is then given by (C.2)where i runs over all N nodes in . This metric depends on the considered volume, so that values are strictly comparable only if computed on equal volumes.

© ESO, 2013