Appendix A: A gauge in which magnetic helicity and magnetic helicity flux are automatically gauge invariant

The purpose of this appendix is to show that there is a particular gauge for the vector potential of the one-dimensional mean field such that $\int\overline{\vec{A}}\cdot\overline{\vec{B}}\,{\rm d} {}z$ is automatically gauge invariant. This allows us then to define a local helicity flux such that its integral over a closed surface (top and bottom of a slab) equals the gauge invariant integrated helicity flux of BD2001.

The evolution equation of the mean (one-dimensional) magnetic vector potential is

$$\dot{\overline{\vec{A}}}=-(\overline{\vec{E}}-\overline{\vec{E}}_0),$$ (A.1)

where $\vec{E}=\eta\mu_0\vec{J}-\vec{u}\times\vec{B}$ is the electric field, $\overline{\vec{E}}_0=\overline{\vec{E}}_0(t)$ is an integration constant, and $\overline{\vec{A}}$ and $\overline{\vec{E}}$ depend only on z and t. From Eq. (A.1) follows the evolution equation for the magnetic helicity density,

$${\partial\over\partial t}({\overline{\vec{A}}}\cdot{\overline... ...rline{\vec{A}}}] =-2\overline{\vec{E}}\cdot\overline{\vec{B}}.$$ (A.2)

In BD2001 the gauge independent magnetic helicity of the mean field was found to be

$$H_{\rm mean}=\int_{z_1}^{z_2} \overline{\vec{A}}\cdot\overlin... ...\vec{z}}\cdot(\overline{\vec{A}}_1\times\overline{\vec{A}}_2),$$ (A.3)

where $\overline{\vec{A}}_1$ and $\overline{\vec{A}}_2$ are the values of $\overline{\vec{A}}$ at z=z₁ and z₂, respectively. At the initial time one can always subtract a constant from $\overline{\vec{A}}$ such that the second term vanishes. This constant turns out to be the average of $\overline{\vec{A}}_1$ and $\overline{\vec{A}}_2$, so we replace initially

$$\overline{\vec{A}}\rightarrow\overline{\vec{A}}-\overline{\ve... ...tyle{1\over2}}(\overline{\vec{A}}_1+\overline{\vec{A}}_2)\cdot$$ (A.4)

Next we choose $\overline{\vec{E}}_0$ such that $\overline{\vec{A}}_1+\overline{\vec{A}}_2$ remains zero at all later times. This yields

$$\overline{\vec{E}}_0={\textstyle{1\over2}}(\overline{\vec{E}}_1+\overline{\vec{E}}_2)\cdot$$ (A.5)

We can then express the two integrated fluxes on z₂ as

$$Q_{\rm mean}^{(2)}=\hat{\vec{z}}\cdot[ (\overline{\vec{E}}_2+... ...over2}}\overline{\vec{E}}_1)\times{\overline{\vec{A}}}_2]\cdot$$ (A.6)

Using the fact that $\overline{\vec{A}}_1+\overline{\vec{A}}_2=0$ we can write

$${\textstyle\int_{z_1}^{z_2}}\overline{\vec{B}}\,{\rm d} {}z =... ...verline{\vec{A}}_2 =-2\hat{\vec{z}}\times\overline{\vec{A}}_1.$$ (A.7)

This allows us to express $Q_{\rm mean}^{(2)}$ as

$$Q_{\rm mean}^{(2)}=-({\textstyle{3\over4}}\overline{\vec{E}}_... ...ot{\textstyle\int_{z_1}^{z_2}}\overline{\vec{B}}\,{\rm d} {}z.$$ (A.8)

At z=z₁ we count the flux as negative when helicity leaves the domain in the downward direction, so

$$Q_{\rm mean}^{(1)}=+({\textstyle{3\over4}}\overline{\vec{E}}_... ...ot{\textstyle\int_{z_1}^{z_2}}\overline{\vec{B}}\,{\rm d} {}z,$$ (A.9)

with $Q_{\rm mean}=Q_{\rm mean}^{(2)}-Q_{\rm mean}^{(1)}$ being the gauge invariant magnetic helicity flux of BD2001. The average upward flux of mean magnetic helicity on the two boundaries is

$$Q_{\rm mean}^{\rm (up)} ={\textstyle{1\over2}}\left[Q_{\rm me... ...{\textstyle\int_{z_1}^{z_2}}\overline{\vec{B}}\,{\rm d} {}z.\;$$ (A.10)