Appendix B: Magnetic helicity in sub-domains

In a periodic domain the sum of the magnetic helicities of any two sub-domains is equal to the magnetic helicity of the entire (periodic) domain. In the non-periodic case this is not the case if the gauge invariant magnetic helicity of BD2001 is used also for the sub-domains. We therefore calculate the magnetic helicity of sub-domains by using the gauge discussed in Appendix A, so the magnetic helicity between the points $z_{\it a}$ and $z_{\it b}$ is then

$$H_{\rm mean}^{\it (ab)}=\int_{z_{\it a}}^{z_{\it b}} (\overli... ...}}-\overline{\vec{A}}_0) \cdot\overline{\vec{B}}\,{\rm d} {}z,$$ (B.1)

where $\overline{\vec{A}}_0={\textstyle{1\over2}}(\overline{\vec{A}}_1+ \overline{\vec{A}}_2)$ is independent of the values of $z_{\it a}$ and $z_{\it b}$. For $z_{\it a}=z_1$ and $z_{\it b}=z_2$we recover Eq. (A.3), and the sum of the magnetic helicities of sub-domains agrees with the magnetic helicity of the whole domain from z₁ to z₂.

Similar to Eq. (A.2), we can now derive an evolution equation for $(\overline{\vec{A}}-\overline{\vec{A}}_0)\cdot\overline{\vec{B}}$. The flux term is then like in Eq. (A.2), but with $\overline{\vec{A}}$being replaced by $\overline{\vec{A}}-\overline{\vec{A}}_0$. The magnetic helicity flux out of an individual sub-domain is then $Q_{\rm mean}^{\it (ab)} =Q_{\rm mean}^{\it (b)}-Q_{\rm mean}^{\it (a)}$, where

$$Q_{\rm mean}^{(\alpha)}=\hat{\vec{z}}\cdot[ (\overline{\vec{E... ...{\vec{A}}}_{\alpha}-{\overline{\vec{A}}}_0)], \quad\alpha=a,b.$$ (B.2)

Again, the sum of net helicity fluxes out of sub-domains equals the gauge invariant net helicity fluxes, $Q_{\rm mean}$, of the full domain. This formula can be applied to value of $z_\alpha$, in particular to the equator. In that case one obtains the horizontally averaged magnetic helicity flux through the surface $z_\alpha=0$.